diff --git "a/INE5T4oBgHgl3EQfWw9u/content/tmp_files/load_file.txt" "b/INE5T4oBgHgl3EQfWw9u/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/INE5T4oBgHgl3EQfWw9u/content/tmp_files/load_file.txt" @@ -0,0 +1,3026 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf,len=3025 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='05561v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='NT] 13 Jan 2023 LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY CHRISTOPH AISTLEITNER, ISTV´AN BERKES AND ROBERT TICHY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to re- cent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudo- randomness, Diophantine equations, and the subsequence principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin’s resonance theorem for convergence systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Uniform distribution and discrepancy 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Arithmetic effects: Diophantine equations and sums of common divisors 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The central limit theorem for lacunary sequences 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm for lacunary sequences 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Normality and pseudorandomness 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Random sequences 24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The subsequence principle 28 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' New results: Exact criteria for the central limit theorem for subsequences 36 Acknowledgments 49 References 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Introduction The word “lacunary” has its origin in the Latin lacuna (ditch, gap), which is a diminutive form of lacus (lake).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Accordingly, a lacunary sequence is a sequence with gaps, and a lacunary trigonometric sum is a sum of trigonometric functions with gaps between the frequencies of consecutive summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The origin of the theory of lacunary sums might lie in Weierstrass’ famous example of a continuous, nowhere differentiable function (1872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since then the subject has evolved into many very different directions, reflecting for example the emergence of modern measure theory and axiomatic probability theory in the early twentieth century, profound 1 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY developments in harmonic analysis and Diophantine approximation, the establish- ment of ergodic theory as one of the key instruments of number theory, or the interest in notions of pseudo-randomness which are associated with the evolution of theoretical computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Throughout this paper we will be concerned with convergence/divergence properties of infinite trigonometric series ∞ � k=1 ck cos(2πnkx) or ∞ � k=1 ck sin(2πnkx), as well as with the asymptotic order and the distributional behavior of finite trigono- metric sums N � k=1 ck cos(2πnkx) or N � k=1 ck sin(2πnkx) (the latter often in the simple case where ck ≡ 1), and with their generalizations ∞ � k=1 ckf(nkx) and N � k=1 ckf(nkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here (ck)k≥1 is a sequence of coefficients, and (nk)k≥1 is a sequence of positive in- tegers (typically increasing), which satisfies some gap property such as the classical Hadamard gap condition nk+1 nk > q > 1, k ≥ 1, or the “large gap condition” (also called “super-lacunarity property”) nk+1 nk → ∞, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Furthermore, f is a 1-periodic function which is usually assumed to satisfy some reg- ularity properties (such as being of bounded variation, being Lipschitz-continuous, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ), and which for simplicity is usually assumed to be centered such that � 1 0 f(x) dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Early appearances of such lacunary sums include the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Sums of the form �N k=1 f(2kx), where f is an indicator function of a dyadic sub-interval of [0, 1], extended periodically with period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Borel used such sums in 1909 to show that almost all reals are “normal”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' more on this topic is contained in Section 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Uniform distribution of sequences ({nkx})k≥1 in Weyl’s seminar paper of 1916;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' more on this in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here and throughout the paper, we write {·} for the fractional part function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kolmogorov’s theorem on the almost everywhere convergence of lacunary trigonometric series if the sequence of coefficients is square-summable (1924), LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 3 a result related to his Three Series Theorem for the almost sure conver- gence of series of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Later it turned out that Kolmogorov’s convergence theorem for trigonometric series actually remains true without any gap condition whatsoever, a result which was widely be- lieved to be “too good to be true” before being established by Carleson [77] in 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More on this in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Foundational work on the distribution of normalized lacunary trigonometric sums, in particular the central limit theorems of Kac (1946) and Salem and Zygmund (1947), and the laws of the iterated logarithms of Salem and Zyg- mund (1950) and of Erd˝os and G´al (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More on this in Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A fundamental observation is that the unit interval, equipped with Borel sets and Lebesgue measure, forms a probability space, and that consequently a sequence of functions such as (cos(2πnkx))k≥1 or (f(nkx))k≥1 can be viewed as a sequence of random variables over this space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' if f is 1-periodic and if (nk)k≥1 is a sequence of positive integers then these random variables are identically distributed, but in general they are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, under appropriate circumstances the gap condition which is imposed upon (nk)k≥1 can ensure that these random vari- ables have a low degree of stochastic dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Consequently lacunary sums often mimic the behavior of sums of independent and identically distributed random vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This viewpoint was in particular taken by Steinhaus, Kac, and Salem and Zygmund in their fundamental work on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In a particularly striking situa- tion, the dyadic functions considered by Borel actually turn out to be a version of a sequence of Bernoulli random variables which are truly stochastically independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' accordingly, Borel’s result on the normality of almost all reals is nowadays usually read as the historically very first version of the strong law of large numbers in prob- ability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' When taking this probabilistic viewpoint, the theory of lacunary sums could be seen as a particular segment of the much wider field of the theory of weakly dependent random systems in probability theory, which is associated with notions such as mix- ing, martingales, and short-range dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, it should be noted that the precise dependence structure in a lacunary function system (f(nkx))k≥1 is con- trolled by the analytic properties of the function f, in conjunction with arithmetic properties of the sequence (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is precisely this interplay between probabilis- tic, analytic and arithmetic aspects which makes the theory of lacunary sums so interesting, so challenging and so rewarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the following sections we want to illustrate some instances of these phenomena in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Uniform distribution and discrepancy Let (xn)n≥1 be a sequence of real numbers in the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such a sequence is called uniformly distributed modulo one (in short: u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1) if (1) 1 N N � n=1 1A(xn) = λ(A) for all sub-intervals A �� [0, 1] of the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The word “equidistributed” is also used for this property, synonymously with “uniformly distributed modulo one”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In this definition, and in the sequel, 1 denotes an indicator function, and λ denotes Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In informal language, this definition means that a sequence is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 if every interval A asymptotically receives its fair share of elements of the sequence, which is proportional to the length of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that (for example as a consequence of the Glivenko–Cantelli theorem) for a sequence of independent, uniformly (0, 1)-distributed random variables (Un)n≥1 one has 1 N N � n=1 1A(Un) = λ(A) almost surely for all intervals A ⊂ [0, 1], so that in a vague sense uniform distribution of a deter- ministic sequence can be interpreted in the sense that the sequence shows “random” behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' more on this aspect in Section 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Uniform distribution theory can be said to originate with Kronecker’s approximation theorem and with work of Bohl, Sierpi´nski and Weyl on the sequence ({nα})n≥1 for irrational α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, the theory only came into its own with Hermann Weyl’s [222] seminal paper of 1916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Among many other fundamental insights, Weyl realized that Definition (1), which in ear- lier work had only be read in terms of counting points in certain intervals, can be interpreted in a “functional” way and can equivalently be written as (2) lim N→∞ 1 N N � n=1 f(xn) = � 1 0 f(x) dx for all continuous functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This viewpoint suggests that uniformly distributed sequences can be used as quadrature points for numerical integration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in the multi- dimensional setting and together with quantitative error estimates this observation forms the foundation of the so-called Quasi-Monte Carlo integration method, a con- cept which today forms a cornerstone of numerical methods in quantitative finance and other fields of applied mathematics (more on this below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Furthermore, Weyl realized that the indicator functions in (1) or the continuous functions in (2) could also be replaced by complex exponentials, as a consequence of the Weierstrass ap- proximation theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' thus by the famous Weyl Criterion a sequence is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 if and only if lim N→∞ 1 N N � n=1 e2πihxn = 0 LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 5 for all fixed non-zero integers h, thereby tightly connecting uniform distribution the- ory with the theory of exponential sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For the particular sequence ({nα})n≥1 it can be easily seen from the Weyl criterion that this sequence is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 if and only if α ̸∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, for other parametric sequences of the form ({nkα})k≥1 the situation is much more difficult, and in general it is completely impossible to determine whether for some particular value of α the sequence is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It turns out that in a metric sense the situation is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Metric number theory arose after the clarification of the concept of real numbers, the realization that the reals drastically outnumber the integers and the rationals, and the development of modern measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Loosely speaking, the purpose of metric number theory is to determine properties which hold for a set of reals which is “typical” with respect to a certain measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' here “typical” means that the measure of the complement is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the present paper the measure under consideration will always be the Lebesgue measure, and a set of reals will be considered typical if its complement has vanishing Lebesgue measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' however, metric number theory has for example also been intensively studied with respect to the Hausdorff dimension or other fractal measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Returning to Weyl’s results, what he proved in the metric setting is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For every sequence of distinct integers (nk)k≥1, the sequence ({nkα})k≥1 is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 for (Lebesgue-) almost all reals α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In other words, even if we cannot specify the set of α’s for which uniform distribution holds, at least we know that the set of such α’s has full Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is amusing that after formulating the result, Weyl continues to write: Wenn ich nun freilich glaube, daß man den Wert solcher S¨atze, in denen eine unbestimmte Ausnahmemenge vom Maße 0 auftritt, nicht eben hoch einsch¨atzen darf, m¨ochte ich diese Behauptung hier doch kurz begr¨unden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='1 One should recall that Weyl’s paper was written in a time of intense conflict of formalists vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' constructivists (with Weyl favoring the latter ones), and only very briefly after the notion of a set of zero (Lebesgue) measure had been introduced at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Today, Weyl’s theorem is seen as one of the foundational results of metric number theory, together with the work of Borel, Koksma, Khinchin and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While uniform distribution modulo one is a qualitative asymptotic property, it is natural that one is also interested in having a corresponding quantitative concept which applies to finite sequences (or finite truncations of infinite sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such 1Even if I think that the value of theorems, which contain an unspecified exceptional set of measure zero, is not particularly high, I still want to give a short justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY a concept is the discrepancy of a sequence, which is defined by DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN) = sup A⊂[0,1] ����� 1 N N � n=1 1A(xn) − λ(A) ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here the supremum is taken over all sub-intervals A ⊂ [0, 1], and it is easy to see that an infinite sequence (xn)n≥1 is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' mod 1 if and only if the discrepancy DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN) tends to 0 as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' With a slight abuse of notation, we will write throughout the paper DN(xn) = DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN) for the discrepancy of the first N elements of an infinite sequence (xn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From a probabilistic perspective, the discrepancy is a variant of the (two-sided) Kolmogorov–Smirnov statistic, where one tests the empirical distribution of the point set x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN against the uniform distribution on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Without going into details, we note that DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN) can be bounded above in terms of exponential sums by the Erd˝os–Tur´an inequality, and that the error when using x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xN as a set of quadrature points to approximate � 1 0 f(x) dx by 1 N �N n=1 f(xn) can be bounded above by Koksma’s inequality in terms of the variation of f and the discrepancy DN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for details see the monographs [96, 161], which contain all the basic information on uniform distribution theory and discrep- ancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [181] for a discussion of equidistribution and discrepancy from the viewpoint of analytic number theory, and [164, 165, 192] for expositions which put particular emphasis on the numerical analysis aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Weyl’s metric result from above can be written as lim N→∞ DN({nkα}) = 0 for almost all α, for any sequence (nk)k≥1 of distinct itegers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Strikingly, the precise answer to the corresponding quantitative problem is still open more than a hundred years later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is known that for every strictly increasing sequence of integers (nk)k≥1 one has (3) DN({nkα}) = O �(log N)3/2+ε √ N � for almost all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is a result of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Baker [38], who improved earlier results of Cassels [78] and of Erd˝os and Koksma [104] by using Carleson’s celebrated convergence theorem in the form of the Carleson–Hunt inequality [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In his paper Baker wrote that [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ] probably the exponent 3/2 + ε could be replaced by ε [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ] but it turned out that this is not actually the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Instead, Berkes and Philipp [64] constructed an example of an increasing integer sequence (nk)k≥1 for which (4) lim sup N→∞ ��� �N k=1 cos(2πnkx) ��� √N log N = +∞ for almost every x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the Erd˝os–Tur´an inequality this gives a corresponding lower bound for the dis- crepancy, which implies that the optimal exponent of the logarithmic term in an upper bound of the form (3) has to be at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' But the actual size of this opti- mal exponent, one of the most fundamental problems in metric discrepancy theory, LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 7 still remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that for pure cosine-sums �N k=1 cos(2πnkx) it is easily seen that one has a metric upper bound with exponent 1/2 + ε in the logarithmic term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this follows from the orthogonality of the trigonometric system, together with Car- leson’s inequality and the Chebyshev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus, in connection with (4), the optimal upper bound in a metric estimate for pure cosine sums is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For sums �N k=1 f(nkx) with f being a 1-periodic function of bounded variation, the optimal exponent also is 1/2 + ε, but this is a much deeper result than the one for the pure cosine case, and was established only recently in [15, 169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By Koksma’s inequality, an upper bound for the discrepancy implies an upper bound for sums of function values for a (fixed) function of bounded variation, but the opposite is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' So while the case of a fixed function f is solved and is an important test case for the discrepancy, the problem of the discrepancy itself (which requires a supremum over a whole class of test functions) is more involved and remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Arithmetic effects: Diophantine equations and sums of common divisors One of the most classical tools of probability theory is the calculation of expectations, variances, and higher moments of sums of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Due to trigonometric identities such as (5) cos a cos b = cos(a + b) + cos(a − b) 2 , the calculation of moments of sums of trigonometric functions (with integer frequen- cies) reduces to a counting of solutions of certain Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' while the first and second moments � 1 0 N � k=1 cos(2πnkx) dx = 0 and � 1 0 � N � k=1 cos(2πnkx) �2 dx = N 2 are trivial and do not depend on the particular sequence (nk)k≥1 (as long as the elements of the sequence are assumed to be distinct),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' interesting arithmetic effects come into play when one has to compute higher moments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and it can be clearly seen how the presence of a gap condition leads to a behavior of the moments which is similar to that of sums of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, assume that we try to calculate � 1 0 � N � k=1 cos(2πnkx) �m dx for some integer m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By (5) this can be written as a sum 2−m � ± � 1≤k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=',km≤N 1 (±nk1 ± · · · ± nkm = 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here the first sum is meant as a sum over all positive combinations of “+” and “-” signs inside the indicator function at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now assume that, for simplicity, we consider the particular sequence nk = 2k, k ≥ 1, which is a prototypical example 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY of a sequence satisfying the Hadamard gap condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then it is not difficult to see that ±nk1 ± · · · ± nkm = 0 is only possible if the elements of the sum cancel out in a pairwise way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' that is, after a suitable re-ordering of the indices, we need to have k1 = k2, k3 = k4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , km−1 = km, and it only remains to count how many such re-orderings are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The result is a combinatorial quantity, and it is exactly the same that arises when calculating an m-th moment of a sum of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus the moments of the trigonometric lacunary sum converge to those of a suitable Gaussian distri- bution, which gives rise to the classical limit theorems for lacunary trigonometric sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The situation is more delicate if one only has the Hadamard gap condition nk+1/nk > q > 1 rather than exact exponential growth, and again more delicate if one considers a sum of dilated functions � f(nkx) instead of a pure trigonometric sum, but the principle described here is very powerful also in these more general situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a long time this was the key ingredient in most of the proofs of limit theorems for lacunary sums;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see for example [103, 145, 193, 201, 214, 221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A different method is based on the approximation of a lacunary sum by a martingale difference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' here the “almost independent” behavior is not captured by controlling the moments of the sum, but in the fact that later terms of the sum (functions with high frequency) oscillate quickly in small regions where earlier summands (functions with much lower frequency) are essentially constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As far as we can say, this method was first used in the context of lacunary sums by Berkes [53] and, indepen- dently, by Philipp and Stout [196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We will come back to this topic in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Broadly speaking, the “almost independent” behavior of sums of dilated functions breaks down when the lacunarity condition is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Many papers have been de- voted to this effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see in particular [57, 59, 102, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In order to maintain the “almost independent” behavior of the sum, there are two natural routes to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the one hand, one could randomize the construction of the sequence (nk)k≥1, and assume that the undesired effects disappear almost surely with respect to the under- lying probability measure – it turns out that this is a very powerful method, and we will come back to it in Section 7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the other hand, when adapting the view- point that the “almost independence” property is expressed in the small number of solutions of certain Diophantine equations, one could try to compensate the weaker growth assumption by stronger arithmetic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A prominent example of a class of sequences for which the latter approach has been very successfully used are the so-called Hardy–Littlewood–P´olya sequences, which consist of all the elements of the multiplicative semigroup generated by a finite set of primes, sorted in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' These sequences are in several ways a natural analogue of lacunary sequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' note that the sequence (2k)k≥1 actually also falls into this framework by consisting of all elements of the semigroup generated by a single prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such sequences gener- ated by a finite set of primes have attracted the attention of number theorists again and again, a particularly interesting instance being F¨urstenberg’s [126] paper on disjointness in ergodic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is known that Hardy–Littlewood–P´olya sequences (if generated by two or more primes) grow sub-exponentially, and the precise (only LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 9 slightly sub-exponential) growth rates are known (Tijdeman [216]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' What is more striking (and a much deeper fact) is that also the number of solutions of the rel- evant linear Diophantine equations can be bounded efficiently – this is Schmidt’s celebrated Subspace Theorem [207] in a quantitative form such as that of Evertse, Schlickewei and Schmidt [107] or Amoroso and Viada [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By a combination of the (slightly weaker) growth condition with the (strong) arithmetic information avail- able for Hardy–Littlewood–P´olya sequences, much of the machinery that is used for Hadamard lacunary sequences can be rescued for this generalized setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see [194] as well as [19, 65, 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We briefly come back to the case of sums of dilated functions � f(nkx) without the presence of a growth condition on (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We assume for simplicity that � 1 0 f(x) dx = 0, so trivially � 1 0 N � k=1 f(nkx) dx = 0, but already the calculation of the variance (6) � 1 0 � N � k=1 f(nkx) �2 dx is in general quite non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If f(x) = cos(2πx), then one can simply use the orthogonality of the trigonometric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If f is a more general function, then one can still express f by its Fourier series, expand the square and integrate, and thus translate the problem of calculating (6) into a problem of counting the solutions of certain linear Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' When carrying out this approach, one is naturally led to the problem of estimating a certain sum involving greatest common divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, assume that f(x) = {x}−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In this case a classical formula (first stated by Franel and first proved by Landau) asserts that � 1 0 f(mx)f(nx) dx = 1 12 (gcd(m, n))2 mn , and consequently � 1 0 � N � k=1 ({nkx} − 1/2) �2 dx = 1 12 � 1≤k,ℓ≤N (gcd(nk, nℓ))2 nknℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The sum on the right-hand side of this equation is called a GCD sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A similar identity holds for example for the Hurwitz zeta function ζ(1 − α, ·), where � 1 0 ζ(1 − α, {mx})ζ(1 − α, {nx}) dx = 2Γ(α)2 ζ(2α) (2π)2α (gcd(m, n))2α (mn)α for α > 1/2, thus leading to a GCD sum � 1≤k,ℓ≤N (gcd(nk, nℓ))2α (nknℓ)α 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY with parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If f(x) is a general 1-periodic function, then one usually does not obtain such a nice exact representation of the variance of a sum of dilated function values, but typically the variance (6) can be bounded above by a GCD sum, which together with Chebyshev’s inequality and the Borel–Cantelli lemma allows to make a statement on the almost everywhere asymptotic behavior of a sum of dilated func- tion values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This connection between sums of dilated functions and GCD sums is explained in great detail in Chapter 3 in Harman’s monograph on Metric Number Theory [136], where mainly the context of metric Diophantine approximation is treated (see also [127, 155]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Recently this connection has also led to a solution of the problem of the al- most everywhere convergence of series of dilated functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Recall that Carleson’s theorem [77] asserts that the series ∞ � k=1 ck cos(2πnkx) is almost everywhere convergent provided that � k c2 k < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is natural to ask which assumption on the sequence of coefficients (ck)k≥1 is necessary to ensure the almost everywhere convergence of the more general series (7) ∞ � k=1 ckf(nkx), under some regularity assumptions on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Gaposhkin [129, 130] obtained some partial results, but a satisfactory understanding of the problem was only achieved very recently, when the connection with GCD sums was fully understood and optimal upper bounds for such sums were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Exploiting this connection with GCD sums, it was shown in [15, 169] that for 1-periodic f which is of bounded variation on [0, 1] the series (7) is almost everywhere convergent provided that ∞ � k=3 c2 k(log log k)γ < ∞ for some γ > 2, and this result is optimal in the sense that the same assumption with γ = 2 would not be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In [16] it was shown that for the class Cα of 1-periodic square integrable functions f with Fourier coefficients aj, bj satisfying aj = O(j−α), bj = O(j−α) for 1/2 < α < 1, a sharp criterion for the almost everywhere convergence of (7) is that (8) ∞ � k=1 c2 k exp �K(log k)1−α (log log k)α � < ∞ with a suitable K = K(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the case of 1-periodic Lipschitz α functions f, Gaposhkin [130] proved that for α > 1/2, the series (7) converges a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' under � k c2 k < ∞ (just like in the case of Carleson’s theorem) and Berkes [60] showed LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 11 that this result is sharp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for α = 1/2 the exact analogue of Carleson’s theorem is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' No sharp convergence criteria exists in the case 0 < α ≤ 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for sufficient criteria for see Gaposhkin [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [4, 68, 128] for a general discussion and several further results for the convergence of series � k ckf(nkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For general periodic f ∈ L2 the direct connection between the integral (6) and GCD sums breaks down, but upper bounds for (6) as well as for (9) � 1 0 � N � k=1 ckf(nkx) �2 dx can be given in terms of the coefficients ck, of the Fourier coefficients of f, and arithmetic functions such as d(n) = � d|n 1, σs(n) = � d|n ds, or the Erd˝os-Hooley function ∆(n) = supu∈R � d|n,u≤d≤eu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See Koksma [156, 157], Weber [220], and Berkes and Weber [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A typical example (see [220]) is the bound � 1 0 �� k∈H ckf(kx) �2 dx ≤ � ∞ � ν=1 a2 ν∆(ν) � � k∈H c2 kd(k) valid for any set H of disjoint positive integers lying in some interval [er, er+1], r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here ak are the complex Fourier coefficients of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using standard methods, such bounds lead easily to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' convergence criteria for sums � k ckf(kx), see the papers cited above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In Wintner [223] it was proved that if f is a periodic L2 function with Fourier coefficients ak, bk, then the series � k ckf(kx) converges in L2 norm for all coefficient sequences (ck)k≥1 satisfying � k c2 k < ∞ if and only if the functions defined by the Dirichlet series ∞ � k=1 akk−s, ∞ � k=1 bkk−s, are bounded and regular in the half plane ℜ(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' There is also a remarkable con- nection between the maximal order of magnitude of GCD sums with the order of ex- treme values of the Riemann zeta function in the critical strip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see [72, 94, 138, 209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Naturally, estimating the integral (6) provides important information also on the asymptotic behavior of averages (10) 1 N N � k=1 f(nkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the Weyl equidistribution theorem, for any 1-periodic f with bounded variation in (0, 1) we have (11) lim N→∞ 1 N N � k=1 f(kx) = � 1 0 f(x) dx a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY (actually for every irrational x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Khinchin [150] conjectured that (11) holds for every 1-periodic Lebesgue integrable f as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This conjecture remained open for nearly 50 years and was finally disproved by Marstrand [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An example for a periodic integrable f and a sequence (nk)k≥1 of positive integers such that the averages (10) do not converge almost everywhere had already been given earlier by Erd˝os [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the other hand, Koksma [157] proved that (11) holds if f ∈ L2 and the Fourier coefficients ak, bk of f satisfy ∞ � k=1 \uf8eb \uf8ed(a2 k + b2 k) � d|k 1 d \uf8f6 \uf8f8 < ∞, and Berkes and Weber [70] proved that the last condition is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' No similarly sharp criteria are known in the case f ∈ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For further results related to the Khinchin conjecture, see [37, 50, 70, 74, 185].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The central limit theorem for lacunary sequences Salem and Zygmund [201] proved the first central limit theorem (CLT) for lacunary trigonometric sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More specifically, they showed that for any integer sequence (nk)k≥1 satisfying the Hadamard gap condition one has lim N→∞ λ � x ∈ (0, 1) : N � k=1 cos(2πnkx) ≤ t � N/2 � = Φ(t), where Φ denotes the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that � 1 0 � N � k=1 cos(2πnkx) �2 dx = N 2 , so the result above contains the “correct” variance for the limit distribution, exactly as it should also be expected in the truly independent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This result has been significantly strengthened since then;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for example, Philipp and Stout [196] showed that under the Hadamard gap condition the function S(t, x) = � k≤t cos(2πnkx), considered as a stochastic process over the space ([0, 1], B[0, 1], λ), is a small per- turbation of a Wiener process, a characterization which allows to deduce many fine asymptotic results for this sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is also known that the central limit theorem for pure trigonometric lacunary sums remains valid under a slightly weaker gap condition than Hadamard’s: as Erd˝os [102] proved, it is sufficient to assume that nk+1/nk ≥ 1 + ck−α, α < 1/2, while such an assumption with α = 1/2 is not suffi- cient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The whole situation becomes very different when the cosine-function is replaced by a more general 1-periodic function, even if it is such a well-behaved one as a LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 13 trigonometric polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, consider (12) f(x) = cos(2πx) − cos(4πx), nk = 2k, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In this case the lacunary sum is telescoping, and it can be immediately seen that there cannot be a non-trivial limit distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A more delicate example is attrib- uted to Erd˝os and Fortet2, and goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let f(x) = cos(2πx) + cos(4πx), nk = 2k − 1, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then it can be shown that N−1/2 �N k=1 f(nkx) does indeed have a limit distribution, but one which is actually non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, for this example one has lim N→∞ λ � x ∈ (0, 1) : N � k=1 f(nkx) ≤ t � N/2 � = 1 √π � 1 0 � t/2| cos(πs)| −∞ e−u2duds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus the limit distribution in this case is a so-called “variance mixture Gaussian”, which can be seen as a normal distribution whose variance is a function rather than a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This limiting behavior can be explained from the observation that (13) f(nkx) = cos((2k+1 − 2)πx) + cos((2k+2 − 4)πx) and (14) f(nk+1x) = cos((2k+2 − 2)πx) + cos((2k+3 − 4)πx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Combining the second term on the right-hand side of (13) with the first term on the right-hand side of (14) we obtain cos((2k+2 − 4)πx) + cos((2k+2 − 2)πx) = 2 cos(πx) cos((2k+2 − 3)πx), so the whole lacunary sum �N k=1 f(nkx) can essentially be written as 2 cos(πx) multiplied with a pure cosine lacunary sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is exactly what the “variance mixture Gaussian” indicates: the limit distribution is actually that of 2 cos(πx) independently multiplied with a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The failure a of Gaussian central limit theorem in the example above can be seen as a consequence of the fact that the Diophantine equation nk+1 − 2nk = 1 possesses many solutions k for this particular choice of sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Equipped with this observation, one could readily construct similar examples with other trigono- metric polynomials f, and other variance mixture Gaussians as limit distributions, by creating situations where there are many solutions k, ℓ to (15) ank − bnℓ = c 2The Erd˝os–Fortet example is first mentioned in print in a paper of Salem and Zygmund [202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' They mention the example without proof, and write: “This remark is essentially due to Erd˝os.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Later the example was mentioned in a paper of Kac [146], who wrote: “It thus came as a surprise when simultaneously and independently of each other, Erd˝os and Fortet constructed an example showing that the limit [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ] need not be Gaussian”, with a footnote: “In Salem and Zygmund this example is erroneously credited to Erd˝os alone.” No proof is given in Kac’s paper either, but he writes: “Details will be given in [a forthcoming] paper by Erd˝os, Ferrand, Fortet and Kac”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such a joint paper never appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY for some fixed a, b, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, interestingly, a special role is played by such equations when c has the particular value c = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' very roughly speaking, solutions of the equation for c = 0 effect only the limiting variance (in a Gaussian distribution), but not the structure of the limiting distribution itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is visible in a paper of Kac [145], who studied the sequence nk = 2k, k ≥ 1, where indeed the only equations that have many solution are of the form 2mnk − nℓ = 0 for some m (the solutions being ℓ = k + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kac proved that for this sequence and any 1-periodic f of bounded variation and zero mean one has (16) lim N→∞ λ � x ∈ (0, 1) : N � k=1 f(nkx) ≤ tσf √ N � = Φ(t) with a limiting variance σ2 f, provided that (17) σ2 f := � 1 0 f 2(x) dx + 2 ∞ � m=1 � 1 0 f(x)f(2mx) dx ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus in this case the limit distribution is always a Gaussian, and the failure of the trivial example in (12) to produce such a Gaussian limit comes from the fact that the limiting variance is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' These observations show that there is a delicate interplay between arithmetic, an- alytic and probabilistic effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in particular, it is obviously not only the order of growth of (nk)k≥1 which is responsible for the fine probabilistic behavior of a la- cunary sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Takahashi [211] proved a CLT (with pure Gaussian limit) under the assumption that nk+1/nk → ∞, and Gaposhkin [128] proved that a CLT (with pure Gaussian limit) holds when nk+1/nk is an integer for all k, or if nk+1/nk → α for some α such that αr ̸∈ Q, r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (and if additionally the variance does not degenerate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A general framework connecting Diophantine equations and the dis- tribution of lacunary sums was established in Gaposhkin’s profound paper [131], where he proved that a CLT (with pure Gaussian limit) holds if for all fixed positive integers a, b the number of solutions k, ℓ of the Diophantine equation (15) is bounded by a constant which is independent of c (where only c ̸= 0 needs to be considered, provided that the variance does not degenerate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One can check the validity of this general condition for sequences satisfying the assumptions mentioned earlier in this paragraph, such as nk+1/nk → ∞ or nk+1/nk → α for αr ̸∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Finally, an optimal result was established in [13]: For (nk)k≥1 satisfying the Hadamard gap condition, the limit distribution of N−1/2 �N k=1 f(nkx) is Gaussian provided that the number of solutions (k, ℓ) of (15), subject to k, ℓ ≤ N, is of order o(N) (for all fixed a, b, uniformly in c ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If, on the other hand, for some a, b, c the number of solutions is Ω(N), then the CLT generally fails to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If the number of solutions with c = 0 also is of order o(N), then the CLT has the “correct” variance � 1 0 f(x)2dx, in perfect accordance with the independent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Even if the number of solutions is of order Ω(N) for some a, b, c, then the deviation of the distribution of N−1/2 �N k=1 f(nkx) from the Gaussian distribution can be quantified in terms of the ratio “(number of solutions)/N”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This shows for example that while the CLT generally fails in the LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 15 case nk+1/nk → p/q, one obtains an “almost CLT” if both p and q are assumed to be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another example for such an “almost CLT” is when the growth constant in the Hadamard gap condition is assumed to be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See the statement of [13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='3] and the subsequent discussion for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that Gaposhkin’s condition implies that the CLT also holds for all subsequences that are picked out of (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is not the case under the assumptions from [13], where one might be able to extract a subsequence along which the CLT fails (by choosing a subsequence which allows a large number of solutions of the relevant Diophantine equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is interesting that the probabilistic behavior of lacunary sums might change when one passes to a subsequence of the original sequence – this is in clear contrast to the bevahior of sums of independent random variables, where any subsequence of course is independent as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A similar remark holds for permutations of lacunary sums resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' permutations of sums of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' These phenomena have received strong attention during the last years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see for example [17, 18, 19, 21, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To give only one sample result, in [17] the following is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As noted above, the CLT is true for pure trigonometric sums under the Erd˝os gap condition nk+1/nk ≥ 1 + ck−α for some α < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, this is only true for the unpermuted sequence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' sorted in increasing order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If permutations of the sequence are allowed, then this gap condition is not sufficient anymore for the validity of the CLT, as is no other gap conditon weaker than Hadamard’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, for any sequence (εk)k≥1 with εk → 0 there exists a sequence of positive integers satisfying nk+1/nk ≥ 1 + εk, together with a permutation σ : N �→ N, such that the permuted (pure trigonometric) sum N−1/2 �N k=1 cos(2πnσ(k)x) converges in distribution to a non-Gaussian limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One can also construct such examples where the norming sequence N−1/2 has to be replaced by (log N)1/2N−1/2 and the limit is a Cauchy distribution, and examples where no limit distribution exists at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See [17] for details on this particular result, and Chapter 3 of [61] for a detailed discussion of permutation-invariance of limit theorems for lacunary (trigonometric) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We close this section with some further references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For Hadamard lacunary (nk)k≥1, the limit distribution of N1/2DN(nkx) was calculated in [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' under suitable Dio- phantine assumptions it coincides with the Kolmogorov distribution, which is the distribution of the range of a Brownian bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A central limit theorem for Hardy– Littlewood–P´olya sequences was established in [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In [87] the Erd˝os–Fortet ex- ample was revisited from the perspective of ergodic theory, and was interpreted in terms of the limiting behavior of certain modified ergodic sums, and generalized to cases such as expanding maps, group actions, and chaotic dynamical systems under the assumption of multiple decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [86, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The limit distribution of N−1/2 �N k=1 cos(2πnkx) for the special sequence nk = k2, k ≥ 1, was determined by Jurkat and Van Horne in [141, 142, 143], and turned out to have finite moments of order < 4, but not of order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The theory of such sums is closely related to theta sums, and goes back to Hardy and Littlewood [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For further related results, see [80, 108, 219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For non-Gaussian limit distributions of N−1/2 �N k=1 cos(2πnkx) 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY near the Erd˝os gap condition nk+1/nk ≥ 1 + ck−1/2 see [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a multidimensional generalization of Kac’s results see [112, 120], and for a multidimensional generaliza- tion of the CLT for Hardy–Littlewood–P´olya sequences (considering a semi-group generated by powers of matrices instead) see [167, 168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [85, 90] for general- izations of the CLT for Hardy–Littlewood–P´olya sequences to a very general setup of sums over powers of transformations/automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm for lacunary sequences Together with the law of large numbers (LLN) and the central limit theorem (CLT), the law of the iterated logarithm (LIL) is one of the fundamental results of prob- ability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Very roughly speaking, the (strong) law of large numbers says that when scaling by N−1 one has almost sure convergence of a sum of random variables, and the central limit theorem says that when scaling by N−1/2 one has a (Gaussian) limit distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm operates between these two other asymptotic limit theorems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in its simplest form, it says that for a sequence (Xn)n≥1 of centered i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables (under suitable extra assumptions, such as boundedness) one has lim sup N→∞ �N n=1 Xn √2N log log N = σ almost surely, where σ is the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Heuristically, the law of the iterated logarithm identifies the threshold between convergence in distribution and almost sure conver- gence for sums of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' indeed, while �N n=1 Xn √2N log log N converges to 0 in distribution by the CLT, it does not converge to 0 almost surely by the LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The first version of the LIL was given by Khinchin in 1924, and a more general variant was established by Kolmogorov in 1929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that the law of large numbers for trigonometric sums or sums of dilated functions is rather unproblematic: for any sequence of distinct integers (nk)k≥1 one has (18) lim N→∞ 1 N N � k=1 f(nkx) = � 1 0 f(x) dx, as long as one can assume a bit of regularity for f (such as f being a trigonometric polynomial, being Lipschitz-continuous, being of bounded variation on [0, 1], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Only if one is not willing to impose any regularity assumptions upon f the situation becomes quite different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see the remarks at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The most basic law of the iterated logarithm for lacunary systems is (19) lim sup N→∞ �N n=1 cos(2πnkx) √2N log log N = 1 √ 2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' under the Hadamard gap condition on (nk)k≥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this was obtained by Salem and Zygmund (upper bound) [203] and Erd˝os and G´al (lower bound) [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Generally speaking, as often in probability theory the lower bound is more difficult to estab- lish than the upper bound, since the latter can be proved by an application of the LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 17 first Borel–Cantelli lemma (convergence part), while the former is proved by the second Borel–Cantelli lemma (divergence part, which needs some sort of stochastic independence as an extra assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that (19) is a perfect analogue of (18) with the “correct” constant on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As in the case of the CLT, replacing pure trigonometric sums by sums of more general 1-periodic functions makes the situation much more delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As in the previous section, a key role is played by Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, while for the CLT it is crucial that the number of solutions of Diophantine equations “stabilizes” in some way to allow for a limit distribution (albeit a potentially non- Gaussian one), no such property is necessary for the validity of a form of the LIL (since, as noted above, this is defined as a lim sup, not as a lim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Instructive examples are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In all examples, we assume that f is 1-periodic with mean zero and bounded variation on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If nk+1/nk → ∞ as k → ∞, then lim sup N→∞ �N n=1 f(nkx) √2N log log N = �� 1 0 f 2(x) dx �1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If nk = 2k, k ≥ 1, then lim sup N→∞ �N n=1 f(nkx) √2N log log N = σf a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', with σ2 f = � 1 0 f 2(x)dx + 2 ∞ � m=1 � 1 0 f(x)f(2mx) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume that nk+1/nk ≥ q > 1, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a constant C (depending on f and on q) such that (20) lim sup N→∞ �N n=1 f(nkx) √2N log log N ≤ C a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If nk = 2k − 1, k ≥ 1, and if f(x) = cos(2πx) + cos(4πx), then (21) lim sup N→∞ �N n=1 f(nkx) √2N log log N = √ 2| cos(πx)| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The first result in this list (due to Takahashi [213]) is in perfect accordance with the LIL for truly independent random sums, in accordance with the fact that also the CLT holds in the “truly independent” form under the large gap condi- tion nk+1/nk → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The second result is an analogue of Kac’s CLT in Equations (16) and (17): as with the CLT, also the LIL holds for the sequence (2k)k≥1, but the limiting variance deviates from the one in the “truly independent” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that in contrast to the CLT case we now do not need to require that σf ̸= 0 for the validity of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The third result (Takahashi [212]) asserts that there is an upper-bound version of the LIL for lacunary sums (even for sequences where there is no convergence of distributions, and any form of the CLT fails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Finally, the fourth 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY result (the Erd˝os–Fortet example for the LIL instead of the CLT) shows the remark- able fact that the lim sup in the LIL for Hadamard lacunary sums might actually be non-constant – this is very remarkable, and a drastic deviation from what one can typically observe for sequences of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In particular this example shows that under the Hadamard gap condition an upper-bound version of the LIL is in general the best that one can hope for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Not very surprisingly, the source of all these phenomena are (as in the previous section) Diophantine equations such as (15), and their number of solutions within the sequence (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' So in the LIL there is again a complex interplay between probabilistic, analytic and arithmetic aspects which controls the fine asymptotic behavior of lacunary sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In probability theory there is a version of the LIL for the Kolmogorov–Smirnov statistic of an empirical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is called the Chung–Smirnov LIL, and in the special case of a sequence (Xn)n≥1 of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables having uniform distribution on [0, 1] (where the Kolmogorov–Smirnov statistic coincides with the discrepancy) it asserts that lim sup N→∞ NDN(Xn) √2N log log N = 1 2 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here the number 1/2 on the right-hand side arises essentially as the maximal L2 norm (“standard deviation”) of a centered indicator function of an interval A ⊂ [0, 1] (namely the indicator function of an interval of length 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Based on the princi- ple that lacunary sequences tend to “imitate” the behavior of truly independent sequences, it was conjectured that an analogue of the Chung–Smirnov LIL should also hold for the discrepancy of ({nkx})k≥1, where (nk)k≥1 is a Hadamard lacunary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This was known as the Erd˝os–G´al conjecture, and was finally solved by Philipp [193], who proved that for any q > 1 there exists a constant Cq such that for (nk)k≥1 satisfying nk+1/nk ≥ q we have (22) 1 √ 32 ≤ lim sup N→∞ NDN({nkx}) √2N log log N ≤ Cq a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An admissible value of Cq was specified in [193] as Cq = 166/ √ 2 + 664/(√2q − √ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The first inequality in (22) follows from (a complex version of) Koksma’s inequality together with (19), so the novelty is the second inequality (upper bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note also that the upper bound in (22) implies Takahashi’s “upper bound” LIL in (20), again as a consequence of Koksma’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Philipp’s result has been extended and refined into many different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The most precise results, many of which were obtain by Fukuyama, show again a fasci- nating interplay between arithmetic, analytic and probabilistic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As a sample we state the following results (all from Fukuyama’s paper [113]): Let nk = θk, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then (23) lim sup N→∞ NDN({nkx}) √2N log log N LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 19 exists and is constant (almost everywhere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Denoting the value of this lim sup by Σθ, we have: If θr ̸∈ Q for all r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , then Σθ = 1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If r denotes the smallest positive integer such that θr = p/q for some coprime p, q, then 1/2 ≤ Σθ ≤ � (pq + 1)/(pq − 1)/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If θr = p/q as above and both p and q are odd, then Σθ = � (pq + 1)/(pq − 1)/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If θ = 2, then Σθ = √ 42/9 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If θ > 2 is an even integer, then Σθ = � (p + 1)p(p − 2)/(p − 1)3/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If θ = 5/2, then Σθ = √ 22/9 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' All these results were obtained by very delicate calculations involving Fourier anal- ysis and Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The calculations from [113] were continued by the same author and his group in [119, 121, 122, 125], so that now we have a relatively comprehensive picture on the behavior of these lim sup’s in the case when (nk)k≥1 is (exactly) a geometric progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In [7, 8] for general Hadamard lacunary sequences (nk)k≥1 a direct connection was established which links the number of solutions of (15) with the value of the lim sup in the LIL, in the same spirit as this was done before in [13] for the CLT (as de- scribed in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In particular, if the number of solutions of (15) is sufficiently small, then the LIL holds with the constant 1/2 on the right-hand side, exactly as in the truly independent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another interesting observation is that if (nk)k≥1 is Hadamard lacunary with growth factor q > 1, and if Σ denotes the value of the lim sup in (23), then the difference |Σ − 1/2| can be quantified in terms of q and tends to zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' as q → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus there is a smooth transition towards the “truly independent” LIL as the growth factor q increases, and under the large gap condition nk+1/nk → ∞ the value of Σ actually equals 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another remark- able fact is that there exist Hadamard lacunary sequences for which the lim sup in the LIL for the discrepancy is not a constant almost everywhere, but rather a function of x, similar to what happened in (21) for the LIL for � f(nkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In some cases the limit functions in the LIL for the discrepancy can be explicitly calculated, and are “surprisingly exotic” (in the words of Ben Green’s MathSciNet review of [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As noted above, Philipp’s LIL for the discrepancy has been extended into many different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, while it is known that the result can fail as soon as the Hadamard gap condition is relaxed to any sub-exponential growth condition, it turns out to be possible to obtain an LIL for the discrepancy when a weaker growth condition is compensated by stronger arithmetic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In particu- lar, an analogue of Philipp’s result has been proved for Hardy–Littlewood–P´olya sequences [195];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see also [5, 65, 123, 215].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As a closing remark concerning the LIL, it is interesting that the optimal value of the lower bound in (22) is still unknown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [25] for more context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 20 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY While much effort has been spent towards understanding the probabilistic behavior of lacunary sums at the scales of the CLT and LIL, it seems that investigations at other scales (such as in particular at the large deviations scale) were only started recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The few results which are currently available point once again towards an intricate connection between probabilistic, analytic and arithmetic effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see the very recent papers [22, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Normality and pseudorandomness Normal numbers were introduced by Borel [73] in 1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From the very beginning the concept of normality of real numbers was associated with “randomness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While normality of real numbers was originally defined in terms of counting the number of blocks of digits, it is not difficult to see3 that a number x is normal in base b if and only if the sequence ({bnx})n≥1 is equidistributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As Borel proved, Lebesgue- almost all real numbers are normal in a (fixed) integer base b ≥ 2, and thus almost all reals are normal in all bases b ≥ 2 (such numbers are called absolutely normal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While normal numbers are ubiquitous from a measure-theoretic perspective,4 it is difficult to construct normal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The most fundamental construction is due to Champernowne, who proved (using combinatorial arguments) that the number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='1 2 3 4 5 6 7 8 9 10 11 12 13 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , which is obtained by a concatenation of the decimal expansions of the integers, is normal in base 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The idea of creating normal numbers by a concatenation of the (b-ary) expansions of the values of (simple) functions at integers (or primes) is still the most popular, and probably most powerful, method in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We only note that Copeland and Erd˝os [89] proved that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='2 3 5 7 11 13 17 19 23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , which is obtained by concatenating the decimal expansions of the primes, is normal in base 10, and refer to [92, 93, 171, 174, 186, 187] for more results of this flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It should be noted, however, that there have been earlier constructions of a con- ceptually very different nature, such as that of Sierpinski [208] in 1917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See [43] for an exposition of Sierpinski’s construction and more context, and see also [44] on an early (unpublished) algorithm of Turing for the construction of normal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We finally mention a very recent idea for the construction of a normal number by Drmota, Mauduit and Rivat [95], which is not based on the concatenation of deci- mal blocks as above, but rather on the evaluation of an automatic sequence along a subsequence of the index set (in this particular case, the Thue–Morse sequences evaluated along the squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see also [183, 210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 3Probably first explicitly mentioned by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Wall in his PhD thesis, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 4Interestingly, while the set of normal numbers is large from a measure-theoretic point of view, it turns out to be small from a topological point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' More precisely, the set of normal numbers is meager (of first Baire category), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the words of Edmund Hlawka [139, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 78]: “Thus whereas the normal numbers almost force themselves on to the measure theorist, the topologist is apt to overlook them entirely.” LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 21 While Lebesgue-almost all numbers are normal and there are some constructions of normal numbers, it is generally considered to be completely hopeless to prove that natural constants such as π, e, √ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' are normal in a given base (although the experimental evidence clearly points in that direction: [189, 218, 224]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Many such open problems “for the next millennium” are contained in Harman’s survey article [137];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see also [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, it is quite clear that the mathematical machin- ery which would be necessary to prove the normality of √ 2 or other such constants is completely lacking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' compare the rather deplorable current state of knowledge on the binary digits of √ 2 as given in [34, 98, 217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A small spark of hope is provided by the very remarkable formulas of Bailey, Borwein and Plouffe (now widely known as BBP formulas), which allow to calculate deep digits of π (and other constants) without the need of computing all previous digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See [51] for a very comprehen- sive “source book” covering computational aspects of π, and see [35] for a very rare example of a possible strategy of what a proof of the normality of π could possibly like (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' also [162]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since normality of x in a base b can be expressed in terms of the equidistribution of the sequence ({bnx})n≥1, it is very natural to consider the discrepancy of DN({bnx}) and call this (with a slight abuse of language) the discrepancy of x (as a normal number, with respect to a base b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Remarkably, it is still unknown how small the discrepancy of a normal number can be (this is known as Korobov’s problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Levin [166] constructed (for given base b) a number x such that DN({bnx}) = O �(log N)2 N � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' by Schmidt’s general lower bound the exponent of the logarithm cannot be reduced below 1, but the optimal size of this exponent remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One of the most interesting, and most difficult, aspects of normal numbers is nor- mality with respect to two or more different bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Extending work of Cassels [79], Schmidt [206] characterized when normality with respect to a certain base im- plies normality with respect to another base, and when this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [48, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, generally speaking it is very difficult to construct numbers which are normal with respect to several different bases, and the “constructions” are much less explicit than the ones of Champernowne and Copeland–Erd˝os men- tioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The problem of the minimal order of the discrepancy of normal num- bers seems to be very difficult when different bases are considered simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Aistleitner, Becher, Scheerer and Slaman [12] constructed a number x such that DN({bnx}) = Ob � N−1/2� for all integer bases b ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this is considered to be an “unexpectedly small” order of the discrepancy by Bugeaud in his MathSciNet review of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is not known if the exponent −1/2 of N in this estimate is optimal or not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' indeed, no non-trivial lower bounds whatsoever (beyond the general lower bound (log N)/N of Schmidt) 22 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY are known for this problem, but it is quite possible that whenever simultaneous nor- mality with respect to different (multiplicatively independent) bases is considered, there must be at least one base for which the discrepancy is “large”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In a recent years, there has been a special focus on algorithmic aspects of the con- struction of normal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A particularly striking contribution was a polynomial- time algorithm for the construction of absolutely normal numbers due to Becher, Heiber and Slaman [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' See also [29, 47, 205].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Related to such algorithmic and computational problems are questions on the complexity of the set of normal num- bers from the viewpoint of descriptive set theory in mathematical logic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in this framework, the rank of the set of normal numbers [152] and absolutely normal num- bers [46] within the Borel hierarchy has been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The notion of normality can be extended in a natural way to many other situa- tions, where it is always understood that normality should be the typical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, one can consider normal continued fractions, where the “expected” number of occurences of each partial quotient is prescribed by the Gauss–Kuzmin measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see for example [1, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Other generalizations consider for example normal- ity with respect to β-expansions [39, 173], a numeration system which generalizes the b-ary expansion to non-integral bases β, or normality with respect to Cantor ex- pansions [3, 109, 175], a numeration system which allows a different set of “digits” at each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a particularly general framework, see [172].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Interestingly, in such generalized numeration systems there can be more than one natural definition of normality, using as starting point for exampe either the idea of counting blocks of digits, or the idea of equidistribution of an associated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The relation be- tween such different (sometimes conflicting) notions of normality has been studied in particular detail for Cantor expansions [2, 170, 176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Normal numbers feature prominently in the chapter on random numbers in Volume 2 of Knuth’s celebrated series on The Art of Computer Programming [153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' There he tries to come to terms with the notion of “random” sequences of numbers, and intro- duces an increasingly restrictive scheme of “randomness” of deterministic sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The concept of normality is also the starting point for one of the (quantitative) mea- sures of pseudorandomness, which were introduced by Mauduit and S´ark¨ozy [179] and then studied in a series of papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note in this context that the transformation T : x �→ bx mod 1, which is at the foundation of the concept of normal numbers, can in some sense be seen as the continuous analogue of the recursive formula which defines a linear congruential generator (LCG), one of the most classical devices for pseudorandom number generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For this connection between normal numbers and pseudo-random number generators, see for example [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another very fruitful aspect of normal numbers is the connection with ergodic theory, which comes from the observation that the sequence ({bnx})n≥1 is the orbit of x under the transforma- tion T from above, and that this transformation is measure-preserving (with respect to the Lebesgue measure) and ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We will not touch upon this connection in LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 23 any detail, and instead refer to [91, 149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another sequence which is often associated with “randomness” is the sequence ({xn})n≥1 for real x > 1, or more generally ({ξxn})n≥1 for ξ ̸= 0 and x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This looks quite similar to a (Hadamard) lacunary sequence such as (bnx)n≥1, but its metric theory is of a very different nature in several respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Both sequences are variants of a geometric progression, but while in the lacunary sequence the base b is fixed and x is assumed to be a “parameter”, now ξ is assumed to be fixed and the base x of the geometric progression is the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While (bnx)n≥1 can in many ways be easily interpreted in terms of harmonic analysis, digital expansions, ergodic theory, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', such simple interpretations fail for ({xn})n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note in particular that in contrast to lacunary sequences there now is no periodicity when replacing x �→ x+1, there is no “orthogonality”, and the calculation of moments of sums � f(xn) does not simply reduce to the counting of solutions of Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Still, what is preserved from the setup of lacunary sequences is that xn (as a function of x) oscillated quickly on intervals where xm is essentially constant, provided that n is significantly larger than m, and there are good reasons to consider systems such as (cos(2πxn))n≥1 to be “quasi-orthogonal” and “almost independent” in some appro- priate sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One of the most fundamental results on this type of sequence is due to Koksma [154]: assuming that ξ ̸= 0 is fixed, the sequence ({ξxn})n≥1 is uniformly distributed mod 1 for almost all x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In particular, when ξ = 1, the sequence ({xn})n≥1 is uniformly distributed mod 1 for almost all x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In very sharp contrast with Koksma’s metric result is the fact that until today not a single example of a number x is known for which ({xn})n≥1 is indeed uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This problem is related with Mahler’s problem on the range of ({(3/2)n})n≥1, which also seems to be completely out of reach for current methods (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [97, 110]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The sequence ({xn})n≥1 is discussed at length in Knuth’s book, where it is conjectured that this sequence is a good candidate to pass several very strict pseudorandomness criteria for almost all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, Knuth conjectured that for all sequences of distinct integers (sn)n≥1 the sequence ({xsn})n≥1 (a subsequence of the original sequence) has a strong equidistribution property called complete uniform distribution, for almost all x > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this was indeed established by Niederreiter and Tichy [188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is also known that ({xn})n≥1 satisfies an law of the iterated logarithm in the “truly independent” form lim sup N→∞ NDN({xn}) √2N log log N = 1 2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', and similarly satisfies a central limit theorem which is perfectly analogous to the one for truly independent systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that Knuth’s assertion that the sequence ({xn})n≥1 shows good pseudo-random behavior for almost all x > 1 is of limited practical use, as long as no such value of x is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The discrete analogue would be to study the pseudo-randomness properties of an mod q for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , where a and q are fixed integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Investigations on the pseudo-randomness properties of such sequence were carried out for example by Arnol’d [32], who experimentally observed 24 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY good pseudorandom behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' also [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To close this section, we note that equidistribution is of course just one property which can be used to characterize “pseudorandom” behavior (essentially by analogy with the Glivenko–Cantelli theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' There are many other statistics which could be applied to a sequence in [0, 1] to determine whether it behaves in a “random” way or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One class of such statistics are gap statistics at the level of the average gap (which is of order 1/N when considering the first N elements of a sequence in [0, 1]), such as the distribution of nearest-neighbor gaps, or the pair correlation statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We do not give formal definitions of these concepts here, but note that they are inspired by investigations of the statistics of quantum energy eigenvalues in the context of the Berry–Tabor conjecture in theoretical physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see [177] for more context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Pseudorandom behavior with respect to such statistics is called “Poisso- nian”, since it agress with the corresponding statistics for the Poisson process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The general principle that lacunary systems show pseudorandom behavior is also valid in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, Rudnick and Zaharescu [199] showed that for (nk)k≥1 satisfying the Hadamard gap condition the sequence ({nkx})k≥1 is Poissonian for almost all x, and Aistleitner, Baker, Technau and Yesha [11] showed that the same holds for ({xn})n≥1 for almost all x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This section on normal numbers and sequences of the form ({ξxn})n≥1 gives of course only a very brief overview of the subject, and has to leave out many interesting aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a much more detailed exposition we refer the reader to the book of Bugeaud [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Random sequences In the previous sections we have illustrated the philosophy that gap sequences ex- hibit many probabilistic properties which are typical for sequences of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In many cases the large gap condition nk+1/nk → ∞ gives “true” random limit theorems, the Hadamard gap condition nk+1/nk ≥ q > 1 is a critical transi- tion point where a mixture of probabilistic, analytic and arithmetic effects comes into play, and the “almost independent” behavior is lost when the gap condition is relaxed below Hadamard’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' There are results which hold under weaker gap condi- tions such as the Erd˝os gap condition nk+1/nk ≥ 1 + ck−α, 0 < α < 1/2, or under additional arithmetic assumptions, but as a whole the Hadamard gap condition is the critical point where the “almost independent” behavior of systems of dilated functions starts to break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, while almost independent behavior is generally lost under a weaker gap condition (without strong arithmetic information), there is another possible per- spective on the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As noted, for a fixed sequence (nk)k≥1 one cannot ex- pect “almost independent” behavior of ({nkx})k≥1, say, without assuming a strong growth condition on (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, even without such a growth condition one might expect that ({nkx})k≥1 shows independent behavior for “typical” sequences (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here the word “typical” of course implies that the sequence has to be taken LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 25 from a generic set in some appropriate space which possesses a measure, so quite naturally this idea leads to considering “random” sequences (nk)k≥1 = (nk(ω))k≥1 which are constructed in a randomized way over some probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Of course there are many possible ways how a random sequence can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From results of Salem and Zygmund [204] for trigonometric sums with random signs it follows easily that if we define a sequence (nk)k≥1 by flipping a coin (independently) for every positive integer to decide whether it should be contained in the sequence or not,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and let P denote the probability measure on the space over which the “coins” are defined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' then for P-almost all sequences as defined above one has (24) 1 √ N N � k=1 cos(2πnkx) D −→ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 1/4) and (25) lim sup N→∞ 1 √2N log log N N � k=1 cos(2πnkx) = 1 2 for almost all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' where N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' σ2) denotes the normal distribution with mean 0 and variance σ2 and D −→ denotes convergence in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that (24) and (25) are not exactly matching with the truly independent case, where the limit distribution would be N(0, 1/2) and the limsup in the LIL would be 1/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The “loss” on the right-hand sides of (24) and (25) comes from the fact that a Dirichlet kernel is “hiding” in this linearly growing sequence, and this kernel is highly localized near 0 and 1 so that its contribution is lost in the CLT and LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the strong law of large numbers (SLLN) clearly nk ∼ 2k as k → ∞, P-almost surely, so the sequences constructed here are very far from satisfying any substantial gap condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in contrast, their (typical) order of growth is only linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It should be noted that the gaps nk+1 − nk in this sequence are not bounded: with full P-probability, nk+1 −nk = 1 for infinitely many k (roughly, in half of the cases), but for infinitely many k, the gap nk+1 − nk has order of magnitude c log k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this follows from the “pure heads” theorem of Erd˝os and R´enyi, see [198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We call an increasing sequence (nk)k≥1 of positive integers a B2 sequence if there exists a constant C > 0 such that for any integer ν > 0 the number of representations of ν in the form ν = nk ±nℓ, k > ℓ ≥ 1, is at most C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By a result of Gaposhkin [131] already mentioned in Section 4, the sequence (f(nkx))k≥1 satisfies the CLT for all Hadamard lacunary (nk)k≥1 and all 1-periodic Lipschitz continuous f if and only if for any m ≥ 1, the set-theoretic union of the sequences (nk)≥1, (2nk)≥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , (mnk)≥1 satisfies the B2 condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='5 No similarly complete result is known for sequences (nk)k≥1 growing slower than exponentially, but Berkes [54] proved that if (nk)k≥1 is 5Note that the definition of the B2 property used in [131] is slightly different from the standard usage in number theory (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [134]) requiring that the number of solutions of ν = nk + nℓ, k > ℓ ≥ 1, is bounded by C, but this does not affect the discussion below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 26 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY a B2 sequence satisfying the gap condition (26) nk+1/nk ≥ 1 + ck−α, k ≥ 1, for some c > 0, α > 0, then (cos(2πnkx))k≥1 satisfies the CLT and LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To verify the B2 property for a concrete sequence (nk)k≥1 is generally a difficult problem, but the situation is quite different for random constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' be disjoint blocks of consecutive integers and let n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' be independent random variables on some probability space (Ω, F, P) such that nk is uniformly distributed over Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly, the number of different sums ±nk1 ± nk2 ± nk3, 1 ≤ k1, k2, k3 ≤ k − 1, is at most 8(k − 1)3, and thus if the size of Ik is ≥ k5, then the probability that nk is equal to any of these sums is ≤ 8k3k−5 = O(k−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus by the Borel-Cantelli lemma, with P-probability 1, such a coincidence can occur only for finitely many k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus the equation ±nk1 ± nk2 ± nk3 ± nk4 = 0, k1 ≤ k2 ≤ k3 < k4 has only finitely many solutions, which implies that (nk)k≥1 is a B2 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let us recall now that by a result of Erd˝os [102], (cos(2πnkx))k≥1 satisfies the CLT with limit distribution N(0, 1/2), provided that (26) holds with α < 1/2, and this result is sharp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a sequence (nk)k≥1 satisfying (26) with α = 1/2 such that the CLT fails for (cos(2πnkx))k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that the counterexample is irregular: while nk+1/nk − 1 is of the order O(k−1/2) for most k, there is also a subsequence along which nk+1/nk → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' One may therefore wonder if regular behavior of nk+1/nk implies the CLT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' in particular, Erd˝os [102] conjectured that the CLT holds for (cos(2πnkx))k≥1 if nk = ⌊e(kβ)⌋ for some β in the range 0 < β ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (Note that for β > 1/2 condition (26) is satisfied with α < 1/2, so the CLT follows from Erd˝os’ result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') This conjecture was proved by Murai [184] for β > 4/9, but for smaller β the problem is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Random constructions provide here important information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kaufman [148] proved that if c is chosen at random, with uniform distribution on a finite interval (a, b) ⊂ (0, ∞), then (cos(2πnkx))k≥1 with nk = e(ckβ) satisfies the CLT with probability 1 for any fixed β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An even wider class of random sequences with the CLT property is obtained by choosing the blocks Ik in the random construction above as the integers in the interval (27) Jk = � e(ckβ)(1 − rk), e(ckβ)(1 + rk) � , rk = o(k−(1−β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A simple calculation shows that these intervals are disjoint for k ≥ k0 and for nk ∈ Jk we have (26) with α = 1 − β, in fact we even have nk+1/nk = 1 + c1(1 + o(1))/k1−β with some constant c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now if rk decreases like a negative power of k, then the length of Jk will be ≥ k5 and thus the constructed random sequence (nk)k≥1 will be a B2 sequence with probability 1, so (cos(2πnkx))k≥1 satisfies the CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In other words, the CLT for (cos(2πnkx))k≥1 holds for a huge class of sequences nk ∼ e(ckβ) for any c > 0, β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 27 Concerning B2 sequences, it is worth pointing out that Erd˝os [101] proved, decades before Carleson’s convergence theorem, that �∞ k=1(ak cos(2πnkx) + bk sin(2πnkx)) is almost everywhere convergent if (nk)k≥1 is a B2 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The question of how slowly B2 sequences can grow is a much investigated problem of number theory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Halberstam and Roth [134], Chapters II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is easily seen that a B2 sequence (nk)k≥1 cannot be o(k2) and Erd˝os and R´enyi [105] proved by a random construction that for any ε > 0 there exists a B2 sequence (nk)k≥1 with nk = O(k2+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Changing the B2 property slightly and requiring that all numbers nk ± nℓ, k > ℓ, are actually different, makes the problem considerably harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The “greedy algorithm” yields a B2 sequence (nk)k≥1 with nk = O(k3), see [180], and it took nearly 40 years to improve this to nk = o(k3), see [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The best currently known (random) construction is due to Ruzsa [200], and satisfies nk = k1/( √ 2−1)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (ωn)n≥1 be a nondecreasing sequence of positive integers tending to +∞ and let us divide the set of positive integers into disjoint blocks I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' such that the cardi- nality of Ik is ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using these blocks in the random construction above, the resulting random sequence (nk)k≥1 cannot be a B2 sequence if (ωn)n≥1 grows slower than any power of n, but it is proved in Berkes [55] that with P-probability 1, (cos(2πnkx))k≥1 still satisfies the CLT and LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The limit distribution here is N(0, 1/2) and the lim- sup in the LIL is 1/ √ 2, so that the “loss of mass” phenomenon observed in the case of the random sequence (nk)k≥1 in the Salem-Zygmund paper [204] does not occur here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The gaps in this sequence satisfy nk+1 − nk ≤ 2ωk+1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' they can grow arbitrarily slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An LIL for the discrepancy of ({nkx})k≥1 under the same gap condition was given in Fukuyama [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In [55] the question was raised if there exists a sequence (nk)k≥1 with bounded gaps nk+1 − nk = O(1) such that the CLT holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Bobkov and G¨otze [71] showed that if we want no loss of mass in the CLT, the answer is negative: if (nk)k≥1 is any increasing sequence of positive integers with nk+1 − nk ≤ L, k ≥ 1, such that N−1/2 �N k=1 cos(2πnkx) has a Gaussian limit distribution N(0, σ2), then necessarily σ2 < 1/2 and L ≥ 1/(1 − 2σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the other hand, Fukuyama [115, 116, 117] showed that for any σ2 < 1/2 there exists indeed a random subsequence (cos(2πnkx))k≥1 of the trigonometric system satisfying the CLT with limit distribution N(0, σ2) and with bounded gaps nk+1 − nk ≤ L with L ∼ 4/(1 − 2σ2) as σ2 → 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This shows that the result of Bobkov and G¨otze is optimal up to a factor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This remarkable result is the “small gaps” counterpart of Erd˝os’ central limit theorem [102]: the latter determines the smallest gap sizes in (nk)k≥1 implying the CLT for (cos(2πnkx))k≥1, while Fukuyama’s result determines the smallest gap size which still allows a CLT with limit distribution N(0, σ2) to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is worth pointing out that the bounded gap sequences in [115, 116, 117] are obtained by rather complicated random constructions, while using the previously discussed simple construction and choosing the nk as independent random variables uniformly distributed over adjoining blocks Ik with equal length results in a random 28 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY sequence (nk)k≥1 satisfying almost surely (28) 1 √ N N � k=1 cos(2πnkx) D −→ N(0, Y ), where Y ≥ 0 is a random variable and N(0, Y ) is a “variance mixture” normal distribution with characteristic function E exp(−Y t2/2), see [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We also note that there is generally no “loss of mass” phenomenon for the LIL for trigonometric series with bounded gaps, see [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For further results for trigonometric series with bounded/random gaps, see [42, 41, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The subsequence principle The purpose of the previous sections was to illustrate the principle that thin subse- quences of the trigonometric system, or thin subsequences of a more general system of dilated functions, exhibit properties which are typical for sequences of indepen- dent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, an analogous principle holds in a much wider framework: it is known that, under suitable technical assumptions, sufficiently thin subsequences of general systems of random variables behave like genuine indepen- dent sequences, in the sense that a general sequence of random variables allows to extract a subsequence showing independent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, Gaposhkin [128, 132] and Chatterji [83, 84] proved that if (Xn)n≥1 is any sequence of random variables satisfying supn EX2 n < ∞, then there exist a subsequence (Xnk)k≥1 and random variables X ∈ L2, Y ∈ L1, Y ≥ 0, such that (29) 1 √ N � k≤N (Xnk − X) D −→ N(0, Y ) and (30) lim sup N→∞ 1 √2N log log N � k≤N (Xnk − X) = Y 1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', where as at the end of the previous section N(0, Y ) denotes the “variance mix- ture”normal distribution with characteristic function E exp(−Y t2/2), and where again D −→ denotes convergence in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A functional (Strassen type) ver- sion of (30) was proved by Berkes [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By a result of Koml´os [159], from any sequence (Xn)n≥1 of random variables satisfying supn E|Xn| < ∞ one can select a subsequence (Xnk)k≥1 such that (31) lim N→∞ 1 N � k≤N Xnk = X a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for some X ∈ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Chatterji [81] proved that if (Xn)n≥1 is a sequence of random vari- ables satisfying supn E|Xn|p < ∞ for some 0 < p < 2, then there exist a subsequence LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 29 (Xnk)k≥1 and a random variable X with E|X|p < ∞ such that (32) lim N→∞ 1 N1/p � k≤N (Xnk − X) = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' These results establish the analogues of the central limit theorem (CLT), the law of the iterated logarithm (LIL), the strong law of large numbers (SLLN) and Mar- czinkiewicz’ strong law for subsequences (Xnk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note the mixed (or randomized) character of (29)–(32): the limit X in the strong law of large numbers, the cen- tering factor X in Marczinkiewicz’ strong law, and the limiting variance Y in the CLT (which also determines the limsup in the LIL) all become random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For fur- ther limit theorems for subsequences of arbitrary random variable sequences, see Gaposhkin [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the basis of these and several other examples, Chatterji [82] formulated the following heuristic principle: Subsequence Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let T be a probability limit theorem valid for all se- quences of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables belonging to an integrability class L defined by the finiteness of a norm ∥ ·∥L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then if (Xn)n≥1 is an arbitrary (dependent) sequence of random variables satisfying supn ∥Xn∥L < +∞ then there exists a subsequence (Xnk)k≥1 satisfying T in a mixed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In a profound study, Aldous [27] proved the validity of the subsequence principle for all distributional and almost sure limit theorems subject to minor technical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To formulate his results, let M denote the class of probability measures on the Borel sets of R, equipped with the L´evy metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By [27], a subset A ⊂ M×R∞ is called a limit statute if: (a) P((λ, X1(ω), X2(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A) = 1 provided X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random vari- ables with distribution λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (b) (λ, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A and � |xi − x′ i| < ∞ implies that (λ, x′ 1, x′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' An a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorem can thus be identified with a limit statute, where the analytic statement of the theorem is expressed by (a), while relation (b) means that a small perturbation of the sequence X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' does not change the validity of the limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let us give two examples of limit statutes representing the strong law of large numbers and the law of the iterated logarithm: A1 = � (λ, x) ∈ A : limN→∞ N−1 �N k=1 xk = |λ|1 � ∪ {(λ, x) : |λ|1 = ∞}, A2 = � (λ, x) ∈ A : lim supN→∞(2N log log N)−1/2 ��N k=1 xk − N|λ|1 � = |λ|2 � ∪ {(λ, x) : |λ|2 = ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here |λ|1 and |λ|2 denote the mean and variance of λ provided they are finite, and we write |λ|1 = ∞, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' |λ|2 = ∞ if � R |x|dλ(x) = ∞, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' � R |x|2dλ(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 30 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY On the other hand, by the definitions in [27], a weak limit theorem for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables is a system T = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , {Gλ, λ ∈ M0}) where (a) M0 is a measurable subset of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (b) For each k ≥ 1, fk = fk(λ, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') is a real function on M × R∞, measurable in the product topology, satisfying the smoothness condition |fk(λ, x) − fk(λ, x′)| ≤ ∞ � k=1 ck,i|xi − x′ i| where 0 ≤ ck,i ≤ 1 and limk→∞ ck,i = 0 for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (c) For each λ ∈ M0, Gλ is a probability distribution on the real line such that the map λ → Gλ is measurable (with respect to the Borel σ-field in M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (d) If λ ∈ M0 and X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' are independent random variables with common distribution λ then fk(λ, X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , ) D −→ Gλ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, the central limit theorem corresponds to the case when M0 is the class of distributions with mean 0 and finite variance, (33) fk(λ, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') = x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' + xk − kE(λ) √ k and Gλ = N(0, Var(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let now (Xn)n≥1 be a sequence of random variables with supn ∥Xn∥L < ∞ with any norm ∥ · ∥L on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then (Xn)n≥1 is bounded in probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' lim K→∞ P(|Xn| > K) = 0 uniformly in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By an extension of the Helly–Bray theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [66]), (Xn)n≥1 has a subsequence (Xnk)k≥1 having a limit distribution conditionally on any event in the probability space with positive probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for any A ⊂ Ω with P(A) > 0 there exists a distribution function FA such that lim k→∞ P(Xnk ≤ t | A) = FA(t) for all continuity points t of FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' According to the terminology of [66], such a subsequence is called determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus when investigating asymptotic properties of sufficiently thin subsequences of sequences (Xn)n≥1 with bounded norms, we can assume, without loss of generality, that (Xn)n≥1 itself is determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As is shown in [27, 66], for any determining sequence (Xn)n≥1 there exists a random measure µ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' a measurable map from the underlying probability space (Ω, F, P) to M) such that for any A with P(A) > 0 and any continuity point t of FA we have (34) FA(t) = EA(µ(−∞, t]) LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 31 where EA denotes conditional expectation given A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This measure µ is called the limit random measure of (Xn)n≥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' see Section 9 below for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' With these preparations, we are now in a position to formulate the subsequence theorems of Aldous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem A (Aldous [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence with limit ran- dom measure µ and let A be a limit statute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xnk)k≥1 such that for any further subsequence (Xmk)k≥1 ⊂ (Xnk)k≥1 we have P((λ(ω), Xm1(ω), Xm2(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem B (Aldous [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence with limit ran- dom measure µ and let T = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , {Gλ, λ ∈ M0}) be a weak limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume that P(µ ∈ M0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a sub- sequence (Xnk)k≥1 such that for any further subsequence (Xmk)k≥1 ⊂ (Xnk)k≥1 we have lim k→∞ P(fk(Xm1(ω), Xm2(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' µ(ω)) ≤ t) = EGµ(ω)(t) at all continuity points t of the distribution function on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Writing out Theorem A and B in the case of the limit statutes A1, A2 above and the weak limit theorem defined by (33), we get the CLT, LIL and SLLN for thin subsequences of determining sequences, as stated in (29), (30), (31) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The proof of Koml´os’ result (31) exemplifies the technique used in the field of sub- sequence behavior before Aldous’ paper [27], and in particular in proving the results (29)–(32) mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As Koml´os showed, if (Xn)n≥1 is a sequence of random variables with bounded L1 norms, then its sufficiently thin subsequences (Xnk)k≥1 are, after a random centering and small perturbation, an identically distributed martingale difference sequence with finite means and thus, by classical martingale theory, they satisfy the SLLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Martingale versions of the CLT and LIL yield also relations (29), (30) and their functional versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' While this method yields several further limit theorems for lacunary sequences, martingale difference sequences cer- tainly do not satisfy all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorems in a randomized form and thus the general subsequence principle cannot be proved in such a way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The proof of Theo- rems A and B in [27] uses a different way and utilizes near exchangeability properties of subsequences of general sequences of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a deter- mining sequence with limit random measure µ and let (Yn)n≥1 be a sequence of random variables, defined on the same probability space as the Xn’s, conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' with respect to µ, with conditional distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (For the construction of such an (Yn)n≥1 one may need to enlarge the probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') Clearly, (Yn)n≥1 is exchangeable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for any permutation σ : N → N of the positive integers, the sequence (Yσ(n))n≥1 has the same distribution as (Yn)n≥1, and it satisfies limit the- orems of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables in a mixed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, if EY 2 1 < ∞ and 32 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY Y = E(Y1 | µ), Z = Var (Y1 | µ), then N−1/2 N � k=1 (Yk − Y ) D −→ N(0, Z) and lim sup N→∞ (2N log log N)−1/2 N � k=1 (Yk − Y ) = Z1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This principle holds in full generality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and distributional limit theorems in the above formalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Indeed, if the Yn are conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' with respect to µ and with conditional distribution µ (a random probability measure on R) and if A is a limit statute, then (35) P((µ, Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A|µ)(ω) = P(µ(ω), Y ∗ 1 , Y ∗ 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' where (Y ∗ n )n≥1 is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' sequence with marginal distribution µ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the definition of limit statute, the last probability in (35) equals 1 and taking expectations we get P((µ, Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') ∈ A) = 1, which is exactly our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Specializing to the case of the limit statutes A1 and A2 above, we get relations (29) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A similar argument works for distributional limit theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now, as is shown in [27], for every k ≥ 1 we have (36) (Xn1, Xn2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Xnk) D −→ (Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Yk) as n1 < n2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' < nk, n1 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In other words, for large indices the finite dimensional distributions of the sequence (Xnk)k≥1 are close to those of the limiting exchangeable sequence (Yk)k≥1 and thus one may expect that limit theorems of (Yk)k≥1 (which, as we have just seen, are mixed versions of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorems) continue to hold for sufficiently thin subse- quences (Xnk)k≥1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Of course, a limit theorem for (Xnk)k≥1 can describe a complicated analytic property of the infinite vector (Xn1, Xn2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Xnk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') which does not follow from the weak convergence of the finite dimensional distributions of the sequence, but with a suitable thinning procedure and delicate analytic ar- guments, Aldous showed an infinite dimensional extension of (36), leading to the validity of Theorems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Although the theorems of Aldous are of exceptional generality, there are important results for lacunary sequences which are not covered by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As was shown by Gaposhkin [128],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for every uniformly bounded sequence (Xn)n≥1 of random variables there exists a subsequence (Xnk)k≥1 and bounded random variables X and Y ≥ 0 such that for any numerical sequence (an)n≥1 satisfying (37) AN := N � k=1 a2 k → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' aN = o(A1/2 N ) LACUNARY SEQUENCES IN ANALYSIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' PROBABILITY AND NUMBER THEORY 33 we have (38) 1 AN N � k=1 ak(Xnk − X) D −→ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and if the second relation of (37) is replaced by (39) aN = o(AN/(log log AN)1/2) then we have (40) lim sup N→∞ 1 � 2A2 N log log AN N � k=1 ak(Xnk − X) = Y 1/2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The difference of these results from (29) and (30) is that in the CLT and LIL we have weighted sums �N k=1 ak(Xnk −X) instead of ordinary sums �N k=1(Xnk −X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For ev- ery fixed coefficient sequence (an)n≥1 the CLT and LIL in (38) and (39) follow from Theorems A and B, but the subsequence (Xnk)k≥1 provided by the proofs depends on (ak)k≥1 and it is not clear that we can select a subsequence (Xnk)k≥1 satisfying (38) and (39) simultaneously for all considered coefficient sequences (ak)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Another important situation not covered by Aldous’ general theorems is when we investigate permutation-invariance of limit theorems for subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since the asymptotic properties of an exchangeable sequence (Yn)n≥1 do not change after any permutation of its terms, it is natural to expect that the conclusions in Theorem A and B remain valid after an arbitrary permutation of the subsequence (Xnk)k≥1 in the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' However, the proofs of Theorem A and B are not permutation- invariant and it does not follow that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', any sequence (Xn)n≥1 of random variables with bounded L1 norms contains a subsequence (Xnk)k≥1 satisfying the strong law of large numbers after any permutation of its terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using ad hoc methods, the latter result has been proved by Berkes [56] and another classical case, namely the unconditional a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' convergence of series � ck(Xnk − X) under � c2 k < ∞ for subse- quences (Xnk)k≥1 of L2 bounded sequences (Xn)n≥1, has been settled by Koml´os [160] (see [27] for another proof via exchangeability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It clearly would be desirable to pro- vide further general results in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We now formulate some structure theorems for lacunary sequences enabling one to handle problems of the kind discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Recall that if (Xn)n≥1 is a determin- ing sequence with limit random measure µ and (Yn)n≥1 is a sequence conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' with respect to the σ-algebra generated by µ and with conditional marginal distributions µ, then there exists a subsequence (Xnk)k≥1 such that (36) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This shows that, in some sense, for large indices the sequence (Xnk)k≥1 resembles the sequence (Yk)k≥1, but this property is far too weak to deduce limit theorems for (Xnk)k≥1 from those valid for the exchangeable sequence (Yk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The following theorem, proved by Berkes and P´eter [63], shows that with a suitable choice of the subsequence (nk)k≥1, the variables (Xnk)k≥1 can be chosen to be close to the Yk in a pointwise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We call a sequence (Xn)n≥1 of random variables ε-exchangeable 34 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY if on the same probability space there exists an exchangeable sequence (Yn)n≥1 such that P(|Xn − Yn| ≥ ε) ≤ ε for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then we have Theorem C (Berkes and P´eter [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables bounded in probability, and let (εn)n≥1 be a sequence of positive reals tending to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then, if the underlying probability space is large enough, thee exists a subsequence (Xnk)k≥1 such that, for all l ≥ 1, the sequence Xnl, Xnl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' is εl-exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that Theorem C provides a different approximating exchangeable sequence (Y (l) j )j≥1 for each tail sequence (Xnl, Xnl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ), with termwise approximating error εl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The following theorem describes precisely the structure of the the sequences (Y (l) j )j≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem D (Berkes and P´eter [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence of random variables, and let (εn)n≥1 be a sequence of positive reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xmk)k≥1 and a sequence (Yk)k≥1 of discrete random variables such that (41) P � |Xmk − Yk| ≥ εk � ≤ εk k = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , and for each k > 1 the atoms of the finite σ-field σ{Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Yk−1} can be divided into two classes Γ1 and Γ2 such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Firstly, (42) � B∈Γ1 P(B) ≤ εk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Secondly, for any B ∈ Γ2 there exist PB-independent random variables {Z(B) j , j = k, k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' } defined on B with common distribution function FB such that (43) PB � |Yj − Z(B) j | ≥ εk � ≤ εk, j = k, k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here FB denotes the limit distribution of (Xn)n≥1 relative to B and PB denotes conditional probability given B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We now give applications of Theorem D to the problems discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' First we note that using Theorem D it is a simple exercise to prove, for suitable subsequences of a uniformly bounded sequences (Xn)n≥1, the weighted CLT and LIL in (38), (40) simultaneously for all permitted coefficient sequences (an)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Next we give a permutation-invariant form of Theorem B for distributional limit theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We call the weak limit theorem T = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , S, {Gµ, µ ∈ M0}) regular if there exist sequences pk ≤ qk of positive integers tending to +∞ and a sequence ωk → +∞ such that (i) fk(λ, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=') depends only on λ, xpk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (ii) fk satisfies the Lipschitz condition |fk(λ, xpk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xqk) − fk(λ′, x′ pk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , x′ qk)| ≤ ≤ 1 ωk qk � i=pk |xi − x′ i|α + ̺∗(λ, λ′) LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 35 for some 0 < α ≤ 1, where ̺∗ is a metric on M0 generating the same topology as the Prohorov metric ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For example, the central limit theorem can be formalized by the functions fk(λ, x[k1/4], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , xk) = x[k1/4] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' + xk − kE(λ) √ k , leading to a regular limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note that originally we formalized the CLT with the functions fk in (33) containing all variables x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', but under bounded second moments the first k1/4 terms here are irrelevant and hence we can always switch to the present version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The same procedure applies in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem E (Aistleitner, Berkes and Tichy [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence with limit random measure ˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let T = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , S, {Gµ, µ ∈ M0}) be a regular weak limit theorem and assume that P(˜µ ∈ M0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xnk)k≥1 such that for any permutation (X∗ k)k≥1 of (Xnk)k≥1 we have (44) fk(X∗ 1, X∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , ˜µ) →d � G˜µdP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In case of the CLT formalized above, assuming supn EX2 n < +∞ implies easily that ˜µ has finite variance almost surely, and thus denoting its mean and variance by X and Y , respectively, we see that the integral in (44) is the distribution N(0, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence (44) states in the present case that 1 √ N N � k=1 (X∗ k − X) D −→ N(0, Y ), which is the permutation-invariant form of the CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Concerning a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorems, a permutation-invariant form of the strong law of large numbers for subsequences of an L1-bounded sequence was proved, as already mentioned, in Berkes [56], and a similar argument yields the corresponding result for the LIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' No permutation-invariant version of the general result in Theorem A has been proved in the literature, but there is no need for that, since a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' limit theorems can be reformulated in a distributional form and thus the proof of Theorem B applies with obvious changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For illustration, we give here the reformulation of the LIL: Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables with E|Xn| ≤ 1, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Put Sn = �n i=1 Xi, Sk,l = �l i=k+1 Xi, and L(N) = (2N log log N)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then lim supN→∞ SN/L(N) = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' iff for any ε > 0 there exists a sequence m1 < m2 < �� · of positive integers such that mk ≥ 5k and P � max mk≤j≤mk+1 Sk,j L(j) > 1 + ε � ≤ 2−k, k ≥ k0, and P � max mk≤j≤mk+1 Sk,j L(j) < 1 − ε � ≤ 2−k, k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 36 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY It is worth pointing out that given a sequence (Xn)n≥1 of random variables, find- ing a subsequence (Xnk)k≥1 satisfying the permutation-invariant form of some limit theorem generally requires a much faster growing sequence (nk)k≥1 than to find a subsequence to satisfy the original limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This is a phenomenon which also occurs for lacunary trigonometric sums or lacunary sums of dilated functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' com- pare the last paragraph of Section 4 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In conclusion we note that if (Xn)n≥1 is a sequence of random variables with fi- nite means over the probability space (0, 1) equipped with the Borel σ-algebra and Lebesgue measure such that for all n ≥ 1 and (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , an) ∈ Rn we have (45) C1 � n � k=1 |ak|p �1/p ≤ E ����� n � k=1 akXk ����� ≤ C1 � n � k=1 |ak|p �1/p for some p ≥ 1 and positive constants C1, C2, then the closed subspace of L1(0, 1) spanned by the Xn is isomorphic with the ℓp space (Hilbert space if p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Relation (45) holds, in particular, if the Xn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' symmetric p-stable random variables with p > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' their characteristic function (Fourier transform) is given by exp(−c|t|p) with some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus applying the subsequence principle to the “limit theorem” (45) provides important information on the subspace structure of L1(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using this method, Aldous [28] proved the famous conjecture that every infinite dimensional closed subspace of L1(0, 1) contains an isomorphic copy of ℓp for some 1 ≤ p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' For a further application of this method, see an improvement of the classical theorem of Kadec and Pe�lczy´nski [147] on the subspace structure on Lp, p > 2, in Berkes and Tichy [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' New results: Exact criteria for the central limit theorem for subsequences By the classical resonance theorem of Landau [163], for a real sequence (xn)n≥1 the series �∞ n=1 anxn converges for all (an)n≥1 ∈ ℓp (1 ≤ p ≤ ∞) if and only if (xn)n≥1 ∈ ℓq, where 1/p + 1/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A deep extension of this result to the case of function series was given by Nikishin [190].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We call a sequence (fn)n≥1 of measurable functions on (0, 1) a convergence system in measure for ℓp if for any real sequence (an)n≥1 ∈ ℓp the series �∞ n=1 anfn converges in measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the case p = 2 Nikishin proved the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem G (Nikishin [190, 191]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A function system (fn)n≥1 over (0, 1) is a conver- gence system in measure for ℓ2 if and only if for any ε > 0 there exists a measurable set Aε ⊂ (0, 1) with measure exceeding 1 − ε and a constant Kε > 0 such that for all N ≥ 1 and all (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , aN) ∈ RN we have (46) � Aε � N � n=1 anfn �2 dx ≤ Kε N � n=1 a2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The sufficiency of (46) is obvious, so the essential (and highly remarkable) state- ment is the converse: if a sequence (fn)n≥1 is a convergence system in measure for LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 37 ℓ2, then, except for a subset of (0, 1) with arbitrary small measure, (fn)n≥1 behaves like an orthonormal sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In the previous section we discussed the subsequence principle stating that suffi- ciently thin subsequences of arbitrary sequences of random variables, subject to mild boundedness conditions, satisfy “all” limit theorems for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables in a mixed (randomized) form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A typical special case of this principle is the following result: Theorem H (Gaposhkin [132]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables satisfying (47) sup n EX2 n < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xnk)k≥1 together with random variables X and Y ≥ 0 such that for any further subsequence (Xmk)k≥1 of (Xnk)k≥1 we have (48) 1 √ N N � k=1 (Xmk − X) D −→ N(0, Y ), where N(0, Y ) denotes the “variance mixture” normal distribution with characteris- tic function E exp(−Y t2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If (X2 n)n≥1 is uniformly integrable then by well-known compactness results (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [99]) there exist a subsequence (Xmk)k≥1 and random variables X ∈ L2 and Y ∈ L1/2, Y ≥ 0, such that (49) Xmk → X weakly in L2, (Xmk − X)2 → Y 2 weakly in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='6 As Gaposhkin [128] showed, in this case the random variables X, Y in (48) can be chosen as in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In Theorem H, condition (47) is not necessary: simple examples show (see below) that there exist sequences (Xn)n≥1 of random variables without any finite moments, but having subsequences satisfying (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The purpose of this section is to give nec- essary and sufficient conditions for the existence of subsequences (Xnk)k≥1 satisfying the randomized CLT (48), and it will turn out that our conditions have the same character as Nikishin’s conditions for the existence of a subsequence being a con- vergence system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' “nice” behavior of the sequence on subsets of the probability space with measure as close to 1 as we wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To formulate our results, call a sequence (Xn)n≥1 of random variables nontrivial if it has no subsequence converging with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' It is easily seen that for non-degenerate sequences the random variable Y in Theorem H is almost surely 6A sequence (ξn)n≥1 of random variables in Lp, p ≥ 1, is said to converge weakly to ξ ∈ Lp if E(ξnη) → E(ξη) for any η ∈ Lq, where 1/p + 1/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This type of convergence should not be confused with weak convergence of probability measures and distributions, called generally convergence in distribution, and denoted by D −→.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 38 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY positive and Gaposhkin’s theorem can be rewritten in a form involving a pure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' not mixed) Gaussian limit distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a nontrivial sequence of random variables satisfying (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0 such that for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and for any set A in the probability space with P(A) > 0 we have (50) PA ��N k=1(Xmk − X) Y √ N < t � → Φ(t) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here PA denotes the conditional probability with respect to A, and Φ is the cumulative distribution function of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The nontriviality of (Xn)n≥1 is assumed here to avoid degenerate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If Xnk → X on some set A with positive probability then for any sufficiently thin subsequence (Xmk)k≥1 of (Xnk)k≥1 we have � |Xmk − X| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A, and consequently a−1 N N � k=1 (Xmk − X) → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A for any norming sequence aN → ∞ (random or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since for any sequence (Xn)n≥1 satisfying (47) (and in fact any tight sequence (Xn)n≥1) there is a subsequence (Xnk)k≥1 and a measurable partition A ∪ B of the probability space such that Xnk converges on A and is nontrivial on B, there is no loss of generality in assuming that (Xn)n≥1 is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly, if (Xn)n≥1 satisfies the conclusion of Theorem J, then so does the sequence (Xn + 2−nZ)n≥1 for any a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' finite random variable Z, and thus the assumption (47) is, as stated above, not necessary in Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Below we will give necessary and suf- ficient condition for the CLT for lacunary subsequences of a given sequence (Xn)n≥1 of random variables without any moment assumption on (Xn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To formulate our results, let us note that if all subsequences (Xmk)k≥1 of a sequence (Xn)n≥1 satisfy (50) for some random variables X, Y , then (Xn)n≥1 is bounded in probability (see Lemma 2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As mentioned in the previous section, every sequence (Xn)n≥1 of random variables bounded in probability has a subsequence (Xnk)k≥1 which has a limit distribution relative to every set A of the probability space with P(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Such a sequence was called determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This concept is the same as that of stable convergence, introduced by R´enyi [197];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' our terminology follows that of functional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence in our investigations we can assume without loss of generality that the original sequence (Xn)n≥1 is determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now if (Xn)n≥1 is determining and FA denotes its limit distribution relative to the set A, then, as we noted in the previous section, there exists a random measure µ (called the limit random measure of (Xn)) such that (51) FA(t) = EA(µ(−∞, t]) LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 39 for any continuity point t of FΩ, where EA denotes conditional expectation relative to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let F• denote the distribution function of µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' we shall call it the limit random distribution of (Xn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We can state now our first new theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a nontrivial sequence of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then the following statements are equivalent: A) There exist a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0 such that (50) holds for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and for any set A ⊂ Ω with P(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' B) For every ε > 0 there is a subsequence (Xnk)k≥1 and a set Aε ⊂ Ω with P(Aε) ≥ 1 − ε such that (52) sup k � Aε X2 nkdP < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If (Xn)n≥1 is determining, then two further equivalent statements are: C) We have (53) +∞ � −∞ x2dF•(x) < +∞ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' D) For every ε > 0 there exists a set Aε ⊂ Ω with P(Aε) ≥ 1 − ε such that (54) +∞ � −∞ x2dFAε(x) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Our second new theorem characterizes sequences (Xn)n≥1 for which (50) holds with X ∈ L2, Y ∈ L1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a nontrivial sequence of random variables defined on an atomless probability space (Ω, F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then the following statements are equivalent: A) There exists a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0, X ∈ L2, Y ∈ L1/2 such that (50) holds for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and all sets A ⊂ Ω with P(A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' B) There exists a subsequence (Xnk)k≥1 and sequences (Yk)k≥1, (τk)k≥1 of random variables satisfying (55) Xnk = Yk + τk, where (56) sup k EY 2 k < +∞, � k |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 40 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY If (Xn)n≥1 has a limit distribution F, then a third equivalent statement is: C) We have (57) +∞ � −∞ x2dF(x) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In other words, for the validity of (50) with X ∈ L2, Y ∈ L1/2, assumption (47) is necessary and sufficient after a small perturbation of (Xn)n≥1, and for identically distributed (Xn)n≥1 even this perturbation is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A particularly simple case when X ∈ L2, Y ∈ L1/2 is satisfied is when X, Y are nonrandom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A trivial example showing the difference between condition (D) of Theorem 1 and condition (C) of Theorem 2 is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let {Hk, k ≥ 1} be a partition of the probability space with P(Hk) = 2−k for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , and let (Xn)n≥1 be a sequence of random variables on this space which is conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' given each Hk with mean 0 and variance 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then (Xn)n≥1 is nontrivial, determining and clearly satisfies condition (D) of Theorem 1, but since it is identically distributed (in fact exchangeable) and since EX2 1 = +∞, condition (C) of Theorem 2 is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Some lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The key for the proof of our theorems is a general structure theorem for lacunary sequences which was proved in [63], and which was stated as Theorem D in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Furthermore, we need the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables such that for some ran- dom variables X, Y with Y > 0 and for all subsequences (Xnk)k≥1 we have (58) N� k=1 (Xnk − X) Y √ N D −→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then (Xn)n≥1 is bounded in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly (58) implies that the sequence (XnN − X)/(Y √ N) is bounded in probability as N → ∞, and thus XnN/ √ N is bounded in probability for any sub- sequence (nk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If (Xn)n≥1 were not bounded in probability then one could find a subsequence (mk)k≥1 and a constant c > 0 such that P(|Xmk| ≥ k) ≥ c for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Xmk/ √ k would not be bounded in probability, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a sequence of random variables and assume that for some random variables X and Y > 0 and all sets A ⊂ Ω with P(A) > 0 we have (59) PA \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N� k=1 (Xk − X) Y √ N < t \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 → Φ(t) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 41 Assume further that (59) remains valid if we replace X, Y by some random variables X∗ and Y ∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then X = X∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and Y = Y ∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From the assumption it follows that the sequences N−1/2 N � k=1 (Xk − X) and N−1/2 N � k=1 (Xk − X∗) are bounded in probability, and thus the same holds for their difference √ N(X−X∗), whence X = X∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To prove Y = Y ∗, fix c > 1 and set A = {Y ∗ ≥ cY }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If P(A) > 0 then clearly we cannot have both (59) and the analogous relation with Y replaced by Y ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus P(A) > 0 for all c > 1 whence Y ∗ ≤ Y a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The same argument yields Y ≤ Y ∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Xn be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables with distribution function F and set Sn = X1 + · · · + Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then for any t > 0 we have (60) P(|Sn| ≤ 2t) ≤ A t √n \uf8eb \uf8ec \uf8ed � |x|≤t x2dF(x) − 2 \uf8eb \uf8ec \uf8ed � |x|≤t xdF(x) \uf8f6 \uf8f7 \uf8f8 2\uf8f6 \uf8f7 \uf8f8 −1/2 , provided the difference on the right-hand side is positive and � |x|≤t dF(x) ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Here A is an absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let F ∗ denote the distribution function obtained from F by symmetrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' From a well-known concentration function inequality of Esseen [106, Theorem 2] it follows that the left-hand side of (60) cannot exceed A1 t √n \uf8eb \uf8ec \uf8ed � |x|≤2t x2dF ∗(x) \uf8f6 \uf8f7 \uf8f8 −1/2 , where A1 is an absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence to prove (60) it suffices to show that � |x|≤t dF(x) ≥ 1/2 implies (61) � |x|≤2t x2dF ∗(x) ≥ � |x|≤t x2dF(x) − 2 \uf8eb \uf8ec \uf8ed � |x|≤t xdF(x) \uf8f6 \uf8f7 \uf8f8 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let ξ and η be independent random variables with distribution function F, and set C = {|ξ − η| ≤ 2t}, D = {|ξ| ≤ t, |η| ≤ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then � |x|≤2t x2dF ∗(x) = � C (ξ − η)2dP 42 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY ≥ � D (ξ − η)2dP = 2 � |ξ|≤t ξ2dP · P(|η| ≤ t) − 2 \uf8eb \uf8ec \uf8ed � |ξ|≤t ξdP \uf8f6 \uf8f7 \uf8f8 2 ≥ � |ξ|≤t ξ2dP − 2 \uf8eb \uf8ec \uf8ed � |ξ|≤t ξdP \uf8f6 \uf8f7 \uf8f8 2 , provided P(|η| ≤ t) ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus (61) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Ω, F, P) be an atomless probability space and X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' a se- quence of random variables on (Ω, F, P) with limit distribution F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exist a subsequence (Xnk)k≥1 and sequences (Yk)k≥1 and (τk)k≥1 of random variables on (Ω, F, P) such that Xnk = Yk + τk, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , such that the random variables Yk have distribution function F, and such that � k |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let ( ˆXn)n≥1 be discrete random variables such that P(|Xn − X′ n| ≥ 2−n) ≤ 2−n, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , and denote by Fn the distribution function of Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly Fn → F and thus εn := ̺(Fn, F) → 0, where ̺ denotes the Prohorov distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By a theorem of Strassen [8] there exists a probability measure µn on R2 with marginals Fn and F such that µn((x, y) : |x − y| ≥ εn) ≤ εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let c be a possible value of ˆXn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since the probability space restricted to A = { ˆXn = c} is atomless, there exists a random variable Vn on this space such that PA(Vn < t) = µn((x, y) : x = c, y < t) µn((x, y) : x = c) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Carrying out this construction for all possible values of c in the range of ˆXn, we get a random variable Vn defined on the whole probability space such that the joint distribution of ˆXn and Vn is µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly the distribution of Vn is F and P(| ˆXn − Vn| ≥ εn) ≤ εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Choosing (nk)k≥1 so that εnk ≤ 2−k we get P � | ˆXnk − Vnk| ≥ 2−k� ≤ 2−k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' P � |Xnk − Vnk| ≥ 2 · 2−k� ≤ 2 · 2−k and thus � k |Xnk − Vnk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' by the Borel–Cantelli lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus the de- composition Xnk = Yk + τk, where Yk = Vnk and τk = Xnk − Vnk, satisfies the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Our final two lemmas concern the properties of the limit random distribution of determining sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 43 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (Xn)n≥1 be a determining sequence of random variables with limit random distribution F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then for any set A ⊂ Ω with P(A) > 0 we have (62) EA \uf8eb \uf8ed +∞ � −∞ x2dF•(x) \uf8f6 \uf8f8 = +∞ � −∞ x2dFA(x), in the sense that if one side is finite then the other side is also finite and the two sides are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The statement remains valid if in (62) we replace the interval of integrations by (−t, t), provided t and −t are continuity points of FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This lemma follows easily from (51) by integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let X, X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' be random variables such that both sequences (Xn)n≥1 and (Xn − X)n≥1 are determining;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' let F• and G• denote, respectively, their limit random distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then +∞ � −∞ x2dG•(x) < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' implies +∞ � −∞ x2dF•(x) < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let ε > 0 and choose a set A ⊂ Ω such that P(A) ≥ 1 − ε and on A both X and +∞ � −∞ x2dF•(x) are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let FA and GA denote the limit random distribution of (Xn)n≥1 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (Xn − X)n≥1 relative to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Replacing Xn by Xn + τn where τn → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' clearly does not change the limit distributions FA, GA, F•, G•, and thus by passing to a subsequence and using Lemma 4 we can assume, without loss of generality, that the Xn are identically distributed on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then EAX2 1 = EAX2 2 = · · · = +∞ � −∞ x2dFA(x), where the last integral is finite by the boundedness of +∞ � −∞ x2dF•(x) on A and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By Minkowski’s inequality and the boundedness of X on A it follows that EA((Xn − X)2) is also bounded, and thus Fatou’s lemma implies that +∞ � −∞ x2dGA(x) ≤ lim inf n→∞ EA � (Xn − X)2� < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using Lemma 5 again it follows that +∞ � −∞ x2dG•(x) < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As the measure of A can be chosen arbitrarily close to 1, we get � x2dG•(x) < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof of the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We begin with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Using diag- onalization and Chebyshev’s inequality it follows that if a sequence (Xn)n≥1 satisfies (B), then it has a subsequence bounded in probability and thus also a determining subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By Lemma 1 the same conclusion holds if (Xn)n≥1 satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus 44 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY to prove our theorem it suffices to prove the equivalence of (A), (B), (C), (D) for determining sequences (Xn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In what follows we shall prove the implications (A) =⇒ (C) =⇒ (D) =⇒ (B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' since (B) =⇒ (A) follows easily from Theorem D in the previous section by diagonalization, this will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume that (Xn)n≥1 is a determining sequence satisfying (A), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a subsequence (Xnk)k≥1 and random variables X, Y with Y > 0 such that for any further subsequence (Xmk)k≥1 of (Xnk)k≥1 and any set A ⊂ Ω with P(A) > 0 we have (63) PA \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N� k=1 (Xmk − X) Y √ N < t \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 → Φ(t) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We claim that (Xn)n≥1 satisfies (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly we can assume without loss of generality that (Xnk)k≥1 = (Xk)k≥1 and since (Xn−X)k≥1 contains a determining subsequence, we can assume also that (Xn − X)n≥1 itself is determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Moreover, since (Xn − X)n≥1 satisfies (C) if and only if (Xn)n≥1 does (see Lemma 6), we can assume that X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume indirectly that (Xn)n≥1 does not satisfy (C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a set B ⊂ Ω with P(B) > 0 such that (64) lim t→∞ t � −t x2dF•(x) = +∞ on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then there exists a set B∗ ⊂ B with P(B∗) ≥ P(B)/2 such that on B∗ the random variable Y is bounded and (63) holds uniformly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a constant K > 0 and a numerical sequence Kt → +∞ such that t � −t x2dF•(x) ≥ Kt and Y ≤ K on B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Also, 1 − F•(t) + F•(−t) → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' as t → ∞, and thus we can choose a set B∗∗ ⊂ B∗ with P(B∗∗) ≥ P(B∗)/2 such that on B∗∗ the last convergence relation holds uniformly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' there exists a positive numerical sequence εt ց 0 such that (65) 1 − F•(t) + F•(−t) ≤ εt on B∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We show that there exists a subsequence (Xmk)k≥1 of (Xnk)k≥1 such that (63) fails for A = B∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since our argument will involve the sequence (Xn)n≥1 only on the set B∗∗, in the sequel we can assume, without loss of generality, that B∗∗ = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' That is, we may assume that (65) holds on the whole probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 45 Let C be an arbitrary set in the probability space with P(C) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Integrating (65) and using (51) and Lemma 5 we get (66) t � −t x2dFC(x) ≥ Kt, 1 − FC(t) + FC(−t) ≤ εt, t ∈ H, where H denotes the set of continuity points of FΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Choose t0 ∈ H so large that εt0 ≤ 1/16 and then choose t1 so large that K1/2 t /4 ≥ 2t2 0 for t ≥ t1, t ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then for t ≥ t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' t ∈ H we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' using the second relation of (66),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' ������ t � −t xdFC(x) ������ ≤ 2t2 0 + � t0≤|x|≤t |x|dFC(x) ≤ 2t2 0 + \uf8eb \uf8ec \uf8ed � |x|≥t0 dFC(x) \uf8f6 \uf8f7 \uf8f8 1/2 \uf8eb \uf8ec \uf8ed � |x|≤t x2dFC(x) \uf8f6 \uf8f7 \uf8f8 1/2 ≤ 2t2 0 + 1 4 \uf8eb \uf8ec \uf8ed � |x|≤t x2dFC(x) \uf8f6 \uf8f7 \uf8f8 1/2 ≤ 1 2 \uf8eb \uf8ec \uf8ed � |x|≤t x2dFC(x) \uf8f6 \uf8f7 \uf8f8 1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and thus we proved that for any C ⊂ Ω with P(c) > 0 we have (67) t � −t x2dFC(x) − 2 \uf8eb \uf8ed t � −t xdFC(x) \uf8f6 \uf8f8 2 ≥ 1 2Kt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' t ≥ t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' t ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since (Xn)n≥1 is bounded in probability, there exists a function ψ(x) ր ∞ such that (68) sup n Eψ(Xn) ≤ 1 (see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let (ak)k≥1 be a sequence of integers tending to +∞ so slowly that ak ≤ log k and (69) δk := ak/ψ(k1/4) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Let further (εn)n≥1 tend to 0 so rapidly that εak ≤ 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By Theorem D there exists a subsequence (Xmk)k≥1 and a sequence (Yk)k≥1 of discrete random variables such 46 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY that (41) holds and for each k > 1 the atoms of the finite σ-field σ{Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Yak} can be divided into two classes Γ1 and Γ2 such that (70) � B∈Γ1 P(B) ≤ εak ≤ 2−k and for each B ∈ Γ2 there exist i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' random variables Z(B) ak+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , Z(B) k on B with distribution FB such that (71) PB � |Yj − Z(B) j | ≥ 2−k� ≤ 2−k, j = ak + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We show that (72) P \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N� k=1 Xmk Y √ N < t \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 → Φ(t) for all t cannot hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' this will complete our indirect proof of (A) =⇒ (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Set S(B) ak,k = k � j=ak+1 Z(B) j , B ∈ Γ2, Sak,k = � B∈Γ2 S(B) ak,k 1B, where 1B denotes the indicator function of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By (71), PB ������ k � j=ak+1 Yj − k � j=ak+1 Z(B) j ����� ≥ 1 � ≤ 2−ak, B ∈ Γ2, and thus using (70) we get (73) P ������ k � j=ak+1 Yj − Sak,k ����� ≥ 1 � ≤ 2 · 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By (68), (69) and the Markov inequality we have P ������ ak � j=1 Xmj ����� ≥ akk1/4 � ≤ ak sup 1≤j≤ak P � |xmj| ≥ k1/4� ≤ akψ(k1/4)−1 = δk, which, together with (73) and (41), yields (74) P ������ k � j=1 Xmj − Sak,k ����� ≥ 2akk1/4 � ≤ 3 · 2−ak + δk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS, PROBABILITY AND NUMBER THEORY 47 Applying Lemma 3 to the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' sequence {Z(B) j , ak + 1 ≤ j ≤ k} and using (67) we get PB ������ S(B) ak,k √ k ����� ≤ 1 � ≤ PB � S(B) ak,k √k − ak ≤ 2 � ≤ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' K−1/2 √ k , where the constant is absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus using (70) and Y ≤ K it follows that (75) P ����� Sak,k Y √ k ���� ≤ 1 K � ≤ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' K−1/2 √ k + 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' If (72) were true then by (74) and ak ≤ log k we would also have Sk,ak/Y √ k D −→ N(0, 1), which clearly contradicts (75) for large k, since the right-hand side tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This completes the proof of (A) =⇒ (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The remaining implications (C) =⇒ (D) and (D) =⇒ (B) of Theorem 1 are easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume first that (C) holds, then for any ε > 0 there exists a set A ⊂ Ω with P(A) ≥ 1 − ε and a constant K = Kε such that +∞ � −∞ x2dF•(x) ≤ K on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Integrating the last relation on A and using Lemma 5 we get (52), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (D) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Assume now that (D) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' for any ε > 0 there exists a set A ⊂ Ω with P(A) ≥ 1 − ε such that (52) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Applying Lemma 4 for (Xn)n≥1 on the set A it follows that there exists a subsequence (Xnk)k≥1 and random variables Yk and τk, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , defined on A such that Xnk = Yk + τk, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' , and such that the random variables Yk have distribution FA on A and τk → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Choose a set B ⊂ A with P(B) ≥ 1 − 2ε such that τk → 0 uniformly on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Then clearly (τn)n≥1 is uniformly bounded on B, and further � B Y 2 k dP ≤ � A Y 2 k dP = P(A) +∞ � −∞ x2dFA(x) ≤ +∞ � −∞ x2dFA(x) < +∞ for each k ≥ 1 by the identical distribution of the Yk’s and (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus on B the sequences (Yk)k≥1 and (τk)k≥1 have bounded L2 norms and thus the same holds for 48 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY Xmk = Yk + τk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' sup k � B X2 mkdP < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In view of P(B) ≥ 1 − 2ε this shows that (Xn)n≥1 satisfies statement (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theorem 2 follows from Theorem 1 and a slightly sharper form of Theorems H and J which was proved in [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We already mentioned the fact that in Theorem H the random variables X, Y appearing in (50) actually satisfy X ∈ L2, Y ∈ L1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Moreover, if instead of (47) we make the slightly stronger assumption that the sequence (X2 n)n≥1 is uniformly integrable then by the weak compactness criteria in L1 and L2 it follows that there exists a subsequence (Xnk) and random variables X ∈ L2, Y ∈ L1/2 such that (76) Xnk → X weakly in L2, (Xnk − X)2 → Y 2 weakly in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As is shown in [163], in this case (50) holds with the random variables X, Y de- termined by (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We turn now to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' As in the case of Theorem 1, it suffices to prove the equivalence of statements (A), (B), (C) in the case when (Xn)n≥1 is determining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Also, since replacing Xn by Xn + τn where � |τn| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' does not affect the validity of (50), the conclusion (B) =⇒ (A) of Theorem (2) is contained in the stronger form of Theorem H mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus it suffices to verify the implications (A) =⇒ (C) and (C) =⇒ (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' To prove (A) =⇒ (C) let us assume that (Xn)n≥1 is determining with limit distribution F, and that there exist a subsequence (Xnk)k≥1 and random variables X ∈ L2, Y ∈ L1/2, Y > 0, such that for all subsequences (Xmk)k≥1 of (Xnk)k≥1 and any set A ⊂ Ω with P(A) > 0 equation (50) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' We show +∞ � −∞ x2dF(x) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Clearly we can assume without loss of generality that (Xnk)k≥1 < (Xk)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' By the implication (A) =⇒ (B) =⇒ (D) of Theorem 1 there is a set A ⊂ Ω with P(A) ≥ 1 − ε and a subsequence (Xnk)k≥1 such that (77) sup k � A X2 nkdP < +∞ and � A x2dFA(x) < +∞, where FA is the limit distribution of (Xn)n≥1 on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Applying Lemma 4 for (Xnk)k≥1 on A it follows that there exists a subsequence (Xmk)k≥1 of (Xnk)k≥1 admitting the decomposition (78) Xmk = Yk + τk on A, where the Yk are identically distributed on A with distribution function FA and � |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Being an identically distributed sequence with finite expec- tation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' the sequence (Y 2 k )k≥1 is uniformly integrable on A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and thus by the sharper form of Theorem H mentioned above it follows that there exists a subsequence (Ypk)k≥1 of (Yk)k≥1 such that Ypk → U weakly in L2(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' (Ypk − U)2 → V 2 weakly in L1(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' LACUNARY SEQUENCES IN ANALYSIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' PROBABILITY AND NUMBER THEORY 49 and for any B ⊂ A with P(B) > 0 we have PB ��N k=1(Ypk − U) V √ N < t � → Φ(t) for all t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' where U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' V are random variables such that U ∈ L2(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' V ∈ L1/2(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' V > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus by (78) and � |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A we get PB ��N k=1(Xmpk − U) V √ N < t � → Φ(t) for all t for any B ⊂ A with P(B) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Comparing with (50) and using Lemma 2 we get U = X, V = Y a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' on A, and thus we proved that Ypk → X weakly in L2(A), (Ypk − X)2 → Y 2 weakly in L1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence EAY 2 = lim k→∞ EA(Ypk − X)2 = lim k→∞(EAY 2 pk − 2EAYpkX + EAX2) = lim k→∞ EAY 2 pk − EAX2 = +∞ � −∞ x2dFA(x) − EAX2, (79) where in the last step we used the fact that the Yk’s have distribution FA on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hence (80) P(A)−1 � A Y 2dP = +∞ � −∞ x2dFA(x) − P(A)−1 � A X2dP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Since X ∈ L2(Ω), Y 2 ∈ L1(Ω), the left-hand side of (80) and the second term on the right-hand side approach finite limits as P(A) → 1 and thus � +∞ −∞ x2dFA(x) also converges to a finite limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the other hand, FA → F as P(A) → 1 and thus Fatou’s lemma implies +∞ � −∞ x2dF(x) ≤ lim inf P (A)→1 +∞ � −∞ x2dFA(x) < +∞, proving the implication (A) =⇒ (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Now if (C) holds then by Lemma 4 there exists a subsequence (Xnk)k≥1 permitting the decomposition (55) where � |τk| < +∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' and Yk are identically distributed with distribution F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' since F has finite variance by (C), the first relation of (56) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Thus (Xn)n≥1 satisfies (B) and the proof of Theorem 2 is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Acknowledgments Christoph Aistleitner is supported by the Austrian Science Fund (FWF), projects F- 5512, I-3466, I-4945, I-5554, P-34763, P-35322 and Y-901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Istvan Berkes is supported by Hungarian Foundation NKFI-EPR No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' K-125569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 50 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Berkes, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Tichy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Lacunary sequences and permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In Depen- dence in probability, analysis and number theory, pages 35–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kendrick Press, Heber City, UT, 2010.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the uniform theory of lacunary series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In Number theory—Diophantine problems, uniform distribution and applications, pages 137–167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Springer, Cham, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [62] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Berkes and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Borda.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 113(6):775–828, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [81] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Chatterji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A general strong law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 9:235–245, 1969/70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [82] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Chatterji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Un principe de sous-suites dans la th´eorie des probabilit´es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' In S´eminaire de Probabilit´es, VI (Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Strasbourg, ann´ee universitaire 1970-1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Journ´ees Probabilistes de Strasbourg, 1971), pages 72–89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 258.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Diffusive behavior of ergodic sums over rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 19(2):1950016, 26, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [87] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='-P.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Note on normal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 52:857–860, 1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} 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Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Interscience Publishers, 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [100] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Erd˝os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the strong law of large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Trans.' metadata={'source': 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39, 1943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [102] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Erd˝os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On trigonometric sums with gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Magyar Tud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Mat.' metadata={'source': 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P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Erd¨os and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' G´al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the law of the iterated logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' I, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Nederl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Monatsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 199(1):113-133, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [112] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The central limit theorem for Riesz-Raikov sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theory Related Fields 100(1):57–75, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [113] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm for discrepancies of {θnx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 118(1-2):155–170, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [114] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm for the discrepancies of a permutation of {nkx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 123(1-2):121–125, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [115] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A central limit theorem and metric discrepancy result for sequences with bounded gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Dependence in Probability, Analysis and Number Theory, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 233–246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kendrick Press 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [116] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A central limit theorem to trigonometric series with bounded gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Theory Related Fields 149(1-2):139–148, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [117] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Pure Gaussian limit distributions of trigonometric series with bounded gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Hungar.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 180(4):731–742, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [126] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Furstenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Math.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Bases, lacunary sequences and complemented subspaces in the spaces Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 21:161–176, 1961/62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [148] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kaufman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' On the approximation of lacunary series by Brownian motion.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Kesseb¨ohmer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Munday, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Stratmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Infinite ergodic theory of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' De Gruyter, Berlin, 2016.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 56 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' AISTLEITNER, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' BERKES AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' TICHY [156] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Koksma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' A 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Some properties of trigonometric series whose terms have random signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' 91:245–301, 1954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [205] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Scheerer.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Paris 349:1261–1263, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' [221] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Weiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' The law of the iterated logarithm for lacunary trigonometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE5T4oBgHgl3EQfWw9u/content/2301.05561v1.pdf'} +page_content=' Trans.' metadata={'source': 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