diff --git "a/-NFAT4oBgHgl3EQfqB2B/content/tmp_files/load_file.txt" "b/-NFAT4oBgHgl3EQfqB2B/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/-NFAT4oBgHgl3EQfqB2B/content/tmp_files/load_file.txt" @@ -0,0 +1,1637 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf,len=1636 +page_content='ANALYSIS OF THE SMOOTHLY AMNESIA-REINFORCED MULTIDIMENSIONAL ELEPHANT RANDOM WALK JIAMING CHEN AND LUCILE LAULIN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In this work, we discuss the smoothly amnesia-reinforced multidimensional elephant random walk (MARW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The scaling limit of the MARW is shown to exist in the diffusive, critical and superdiffusive regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We also establish the almost sure convergence in all of the three regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The quadratic strong law is displayed in the diffusive regime as well as in the critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The mean square convergence towards a non-Gaussian random variable is established in the superdiffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Similar results for the barycenter process are also derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Finally, the last two Sections are devoted to a discussion of the convergence velocity of the mean square displacement and the Cram´er moderate deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The amnesia-reinforced elephant random walk 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' A correlated martingale approach 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Scaling limit and convergence 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Scaling limit of the barycenter process 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Velocity of quadratic mean displacement 21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Cram´er moderate deviations 23 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Technical Lemmas 24 References 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Introduction The study of reinforced processes and reinforced random walks has known a growing interest over the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In particular, random walks on graphs, or more precisely edge [37] or vertex [39] reinforced random walks, have been the subject of a great number of contributions, see also [1, 12, 27] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The insight of introducing reinforcement mechanisms to stochastic processes has also shed light on more applied models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In [30], the adaptive strategy of an agent who plays a two-armed bandit machine was described as a self-reinforced random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The philosophy of stochastic reinforcement has also been discussed in the topics of evolutionary ecology [4] and machine learning theory [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Another manifestation of reinforced P´olya urn models on financial economics can be found in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We also refer the readers to [38] for a comprehensive and extensive survey on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The Elephant Random Walk (ERW) is a discrete-time random walk, introduced by Sch¨utz and Trimper [40] in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It was referred to as the ERW in allusion to the traditional saying that elephants can always remember anywhere they have been.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' As it was pointed out [12] by Bertoin 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 60G50, 60G42, 62M09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Reinforced random walk, scaling limit, Cram´er moderate deviation, martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='08644v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='PR] 20 Jan 2023 2 JIAMING CHEN AND LUCILE LAULIN (a) Diffusive regime (b) Critical regime (c) Superdiffusive regime Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The two-dimensional ERW with amnesia (in blue) and its barycenter (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' who relied on K¨ursten’s work [29], the ERW is a special case of step-reinforced random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In fact, the ERW is reinforced because its behavior is influenced by its past : the ERW may have a tendency to do the same thing over and over, or on the contrary, it may try to compensate its previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This different types of behavior, here-called regimes, are ruled by the memory parameter p and it is well-known that the ERW shows three regimes of behavior and that the critical value is p = 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The ERW in dimension d = 1 has received a lot of attention from mathematicians and physicists over the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The almost sure convergence and the asymptotic normality of the position of the ERW were established in the diffusive regime p < 3/4 and the critical regime p = 3/4, see [3, 9, 16] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the superdiffusive regime p > 3/4, Bercu [5] proved that the limit of the position of the ERW is not Gaussian and Kubota and Takei [28] showed that the fluctuation of the ERW around this limit is Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' To obtain those asymptotics, various approaches have been followed : Baur and Bertoin [3] went with the connection to P´olya- type urns while martingales were used by Bercu [5] and Coletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [16] and the construction of random trees with Bernoulli percolation have been explicited by K¨ursten [29] and Businger [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Other quantities of interest regarding the ERW have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For example, Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [20] provided the Cramer moderate deviations associated with the ERW in dimension 1 and, more recently, Hayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [26] studied the rate of quadratic mean displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Bercu and Laulin [9] introduced the multidimensional ERW (MERW), where d ≥ 1, and estab- lished the natural extensions of the results [5] in dimension d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, they investigated the center of mass of the MERW [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In both papers, they extensively used a martingale approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Bertenghi [10] made use of the connection to P´olya-type urns in order to establish functional results for the MERW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Finally, the ERW with changing memory has also been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The ERW with linearly reinforced memory has been studied by Baur [1] via the urn approach, and Laulin [31] using martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Gut and Stadm¨uller [25] proposed an amnesic ERW where the elephant could stop and only remember the first (and second) step it tooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' They also investigated the case where the elephant only remembered a fixed or time-evolving portion of its past (recent or distant) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the recent work [32], Laulin introduced smooth amnesia to the memory of the ERW and established the asymptotic behavior of this new process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The idea of our paper is to generalise the work [32] in dimension 1 to the dimension d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In other words, we introduce smooth amnesia to the memory of the multidimensional elephant random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 40 19 30 20 10 0 10 20 0 10 20 30 40 50 60400 300 200 100 0 0 50 100 150 200 250 35060 50 40 30 20 10 0 0 200 400 600 800MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 3 Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In Section 2, we introduce the basic setting of the elephant random walk (Sn)n∈N placed under an amnesia reinforcement mechanism, which is controlled by the memory sequence (βn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This type of multidimensional reinforced random walked is named as the multidimensional amnesia-reinforced elephant random walk (MARW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Similar to the ERW with the amnesia reinforcement, the MARW also admits a martingale structure, which is discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Unlike the usual ERW, the additional amnesia-reinforcement induces two discrete-time martingales, instead of a single martingale, which are strongly correlated in a nontrivial fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Such strong correlation of martingales will eventually pose some computational difficulties when we analyze the limiting behavior of the MARW in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For instance, when we compute the pointwise limit and the scaling limit of (Sn)n∈N in the diffusive regime, the two strongly correlated martingales have to be dealt with separately, see [8, 31, 32] for the same methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' As a courtesy to our readers, we give a preview of some of our main results, whose proofs will be deferred to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2, and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the diffusive regime, we have the almost sure convergence, 1 nSn → 0 as n → ∞ P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Another logarithmic scaling to the MARW yields the quadratic strong law, 1 log n n � k=1 SkST k k2 → C(p, (βn)n∈N) · 1 dId as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' where the constant C(p, (βn)n∈N) > 0 depends only on the parameter p and the control sequence (βn)n∈N of the amnesia-reinforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Using square-root scaling factor, we observe that the MARW also admits a scaling limit in the diffusive regime, or convergence in distribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that � 1 √nS⌊nt⌋, t ≥ 0 � =⇒ � Wt, t ≥ 0 � where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process such that W0 = 0 and with covariance structure given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It is also of interest to look at the barycenter process (Gn)n∈N of the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Its definition as well as its limiting behavior are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Similar to the discussion of the MARW, we obtain its pointwise convergence, quadratic strong law, and its scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In particular, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5 states that the barycenter process admits a scaling limit at the diffusive regime, or convergence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that � 1 √nG⌊nt⌋, t ≥ 0 � =⇒ � 1 � 0 Wtr dr, t ≥ 0 � where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3 with its covariance structure defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' A natural question to ask is how fast the limiting Theorems in Section 4 are carried on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Section 6 provides a quantitative estimate on the mean square convergence velocity of the pointwise limit, quadratic strong law, and the scaling limit of the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It should be possible to derive similar convergence velocity to the barycenter process, which is not computed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In Section 7, we end this work with a discussion on the Cram´er moderate deviations of the MARW in the diffusive and critical regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' As a preview of our result in this Section, let (ϑn)n∈N ⊆ R be a non-decreasing sequence so that ϑn/√n → 0 as n → ∞, and wn the sequence with asymptotic 4 JIAMING CHEN AND LUCILE LAULIN behavior described in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Take any non-empty Borel set B ⊆ Rd, then we have − inf x∈int B 1 2∥x∥2 ≤ lim inf n→∞ ϑ−2 n log P �anµnSn ϑn√wn ∈ B � ≤ lim sup n→∞ ϑ−2 n log P �anµnSn ϑn√wn ∈ B � ≤ − inf x∈cl B 1 2∥x∥2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) where int B and cl B denote the interior and the closure of B ⊆ Rd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This is the Cram´er moderate deviations for the MARW in the diffusive and critical regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moreover, we chose to postpone some technicalities regarding the analysis of the random walk to the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' That way, the reader can focus on the main Theorems and the ideas of their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' However, some analogous technicalities are displayed in the proof of the Theorems on the barycenter such that the reader can also have a complete overview of the work needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Other probabilistic aspects of interest to the MARW include the statistical inference and an analysis on the Fisher information, see [7], as well as the Wasserstein distance of the reinforced random walk, see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Perturbations of the amnesia intensity and its stability for the MARW is also of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' A similar topic for another type of stochastic process, the Schramn- Loewner evolution, has been considered in [2, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The transience and recurrence property of the MARW remains unknown, to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Readers are referred to [11, 20] for an exposition on the ERW without the amnesia reinforcement mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The amnesia-reinforced elephant random walk To begin with, let us properly introduce the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It is the natural extension to higher dimensions of the one-dimensional MARW, defined in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For an arbitrarily given dimension d ≥ 1, let (Sn)n∈N be a (reinforced) random walk on Zd starting from the origin at time n = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' S0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' At time n = 1, the reinforced random walk moves to one of the 2d nearest-neighbors with equal probability 1/2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' After that, at time n ≥ 1, the reinforced random walk chooses at random an integer 1 ≤ k ≤ n among the past times and performs the same step with probabily p, or goes in any of the 2d − 1 other directions with probability (1 − p)/(2d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This random walk possesses the amnesia property, in the sense that it remembers its most recent past steps better than its remote past steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Colloquially, this random walk has higher probability to choose its recent steps than its earlier steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From a mathematical perspective, the position of this reinforced random walk at time n+1 ≥ 1 is given by Sn+1 = Sn + Xn+1 with Xn+1 being defined as the step of this random walk at time n + 1, satisfying Xn+1 = An+1Xβn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Here An+1 is a random d × d matrix given by P(An = +Id) = p, and, for all 1 ≤ k ≤ d − 1, P(An = −Id) = P(An = +Jk d ) = P(An = −Jk d ) = 1 − p 2d − 1 where Id is the identity matrix of order d, Id = (δi,j)d and Jd = C(0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , 0) is the circulant matrix of order d such that J = (δi+1,j)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It is easy to observe that the fixed permutation matrix Jd satisfied Jd d = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The distribution of the memory βn of the reinforced random walk is such MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 5 that the probability of choosing a fixed past time k ∈ N decays approximately with rate kβ/nβ+1, where β ≥ 0 is the amnesia parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (a) n = 10 (b) n = 100 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Evolution of the distribution of the memory β depending on the value of β and the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' To be precise, this random walk chooses βn+1 according to P � βn+1 = k � = (β + 1)Γ(β + k)Γ(n) Γ(k)Γ(β + n + 1) = β + 1 n µk µn+1 for all 1 ≤ k ≤ n, where µn = n−1 � k=1 � 1 + β k � = Γ(β + n) Γ(n)Γ(β + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) (a) d = 1 (b) d = 2 (c) d = 3 (d) d = 10 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Competition between the dimension and the amnesia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Figure 3 aims to give a better understanding on how amnesia affects the MARW in various dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The horizontal axis corresponds to p (from 0 to 1) and the vertical axis corresponds to β (from 0 to 10, arbitrary chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The diffusive regime, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' when p < 4dβ+2d+1 4d(β+1) or a < 1− 1 2(β+1), is in blue while the superdiffusive regime is in red, see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1 for the definition of the regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' One can observe that when the amnesia parameter β grows, the superdiffusive regime tends to be less represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It should also be noted that when the dimension grows the superdiffusive regime is more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, the amnesia is somehow leading the MARW to a behavior closer to the one in dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' When β vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' β = 0, the MARW reduces to the multidimensional elephant random walk (MERW) introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The two random variables An and βn are constructed to be conditionally independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' At each time n, define the σ-algebra Fn = σ(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then (Fn)n∈N is a discrete-time filtration to which the MARW is clearly adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' β= 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5 β= 1 β= 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4 β= 5 β= 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='0 2 4 9 00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10 β= 0 β= 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='08 β= 2 β= 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='06 β= 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='04- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='00 0 20 40 60 80 10010 8 - 6 4 2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='0 p10 8 - 6 B 4 +0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='0 p10 8 - 6 B 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='0 p10 8 - 6 B 4 +0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='0 p6 JIAMING CHEN AND LUCILE LAULIN Since An and βn are conditionally independent, we clearly have E � Xn+1|Fn � = E � An � E � Xβn+1|Fn � = 2dp − 1 2d − 1 E � n � k=1 Xk1{βn+1=k}|Fn � = 2dp − 1 2d − 1 · β + 1 nµn+1 n � k=1 µkXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) We further denote a = 2dp − 1 2d − 1 and Yn = n � k=1 µkXk (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) such that E � Yn+1|Fn � = � 1 + a(β + 1) n � Yn = γnYn with γn = 1 + a(β + 1)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hereafter, for each n ≥ 1, let an = n−1 � k=1 γ−1 k = Γ(n)Γ(a(β + 1) + 1) Γ(a(β + 1) + n) and wn = n � k=1 (akµk)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) From a Gamma function estimate, also see in [31], we have that na(β+1)an → Γ(a(β + 1) + 1) as n → ∞ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and n−βµn → Γ(β + 1)−1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' A correlated martingale approach Define the following two Rd-valued processes by Mn = anYn and Nn = Sn + a(β + 1) β − a(β + 1)µ−1 n Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The Rd-valued processes (Mn)n∈N and (Nn)n∈N defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) are locally square-integrable martingales adapted to (Fn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since, both Mn and Nn are finite sums for each n ≥ 1, the square-integrability and adapt- ness are immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4), we have E � Mn+1|Fn � = anγ−1 n Yn + anµnγ−1 n E � Xn+1|Fn � = anYn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' And by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2), we have E � Nn+1|Fn � = E � Sn+1 + a(β + 1) β − a(β + 1)µ−1 n+1Yn+1|Fn � = Sn + a(β + 1) β − a(β + 1)µ−1 n Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Notice that via introducing the martingales (Mn)n∈N and (Nn)n∈N, we can write Sn as Sn = Nn − a(β + 1) β − a(β + 1)(anµn)−1Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) This writing is the key on which rely all of our analysis and our martingale approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moreover, the asymptotic behavior of (Mn)n∈N is closely related to wn defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In fact, we have the following asymptotic result, which states the three regimes of the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the diffusive regime when p < 4dβ+2d+1 4d(β+1) or a < 1 − 1 2(β+1), we have wn n1−2(a(β+1)−β) → l(β) as n → ∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 7 with l(β) = 1 1 + 2(β − a(β + 1)) �Γ(a(β + 1) + 1) Γ(β + 1) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the critical regime when p = 4dβ+2d+1 4d(β+1) or a = 1 − 1 2(β+1), we have wn log n → �Γ(β + 1 + 1 2) Γ(β + 1) �2 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) In the superdiffusive regime when p > 4dβ+2d+1 4d(β+1) or a > 1 − 1 2(β+1), we have wn → ∞ � k=1 �Γ(a(β + 1) + 1)Γ(β + k) Γ(a(β + 1) + k)Γ(β + 1) �2 < ∞ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) In order to investigate the asymptotic behavior of (Sn)n∈N, we first introduce an arbitrarily fixed test non-zero vector u ∈ Rd and we define Mn(u) = uT Mn and Nn(u) = uT Nn for each n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It is then clear that (Mn(u))n∈N (Nn(u))n∈N are real-valued locally square-integrable martingales for each fixed u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We further infer that (Sn(u))n∈N satisfies an equation analogous to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In this setting, we have reduced the multidimensional martingales to real-valued martingales without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This technique greatly simplifies our martingale analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From now on, we fix the test vector u ∈ Rd and we introduce the two-dimensional martingale (Ln(u))n∈N defined as Ln(u) = � Nn(u) Mn(u) � for each n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) Denote the martingale increment ϵn+1 = Yn+1 − γnYn for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then (ϵn)n∈N satisfies the martingale difference relation E[ϵn+1|Fn] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We obtain that ∆Ln+1(u) = Ln+1(u) − Ln(u) = � Sn+1(u) − Sn(u) + a(β+1) β−a(β+1) � µ−1 n+1Yn+1(u) − µ−1 n Yn(u) � an+1Yn+1(u) − anYn(u) � = � βµ−1 n+1 β−a(β+1) � µn+1Xn+1(u) − (γn − 1)Yn(u) � an+1ϵn+1(u) � = � βµ−1 n+1 β−a(β+1) an+1 � ϵn+1(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Scaling limit and convergence In this section, we discuss the scaling limit as well as the almost sure convergence in the diffusive, critical and the superdiffusive regimes, depending on the value of p with respect to (4dβ +2d+1)/(4d(β +1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We also give the quadratic strong law in the diffusive regime as well as in the critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Afterwards, the mean square convergence is established in the superdiffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the almost sure convergence 1 nSn → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 8 JIAMING CHEN AND LUCILE LAULIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have from [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15] again that, for all γ > 0, ∥Mn∥2 λmax⟨M⟩n = o �� log Tr⟨M⟩n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) From equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9) and the fact that λmax⟨M⟩n ≤ Tr⟨M⟩n ≤ wn, we get ∥Mn∥2 = o � wn � log wn �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3), we observe ∥Mn∥2 = o � n1−2(a(β+1)−β)� log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since Mn = anYn, we have from equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) ∥Yn∥2 (nµn+1)2 = o � n−1� log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' which implies Yn nµn+1 → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10) and [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15] again, we find that ∥Nn∥2 = o � n � log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) Moreover, we obtain from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) 1 n2 ����Sn + a(β + 1) (β − a(β + 1))µn+1 Yn ���� 2 = o � n−1� log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, we conclude that Sn n + a(β + 1) β − a(β + 1) · Yn nµn+1 → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the quadratic strong law 1 log n n � k=1 SkST k k2 → 2β + 1 − a (1 − a)(1 − 2(a(β + 1) − β)) · 1 dId as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We will check that all the conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3] are satisfied, see also [14, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) is satisfied thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4 while the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) directly follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5 and the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) is exactly the statement of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Therefore, 1 log � det V −1 n �2 n � k=1 �(det Vk)2 − (det Vk+1)2 (det Vk)2 � VkLk(u)Lk(u)T V T k → 1 duT uVt=1 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' On the one hand, we have from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='24) that 1 log n n � k=1 �(det Vk)2 − (det Vk+1)2 (det Vk)2 � VkLk(u)Lk(u)T V T k → 2(1 − a)(β + 1) d uT uVt=1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='24), we have n �(det Vn)2 − (det Vn+1)2 (det Vn)2 � → 2(1 − a)(β + 1) as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Finally, we obtain from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) that 1 log n n � k=1 uT SkST k u k2 = 1 log n n � k=1 vT VkLk(u)Lk(u)T V T k v k → vT Vt=1v · 1 duT u (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 9 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since u ∈ Rd is arbitrary, the assertion follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The MARW admits a scaling limit at the diffusive regime, or convergence in distribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that � 1 √nS⌊nt⌋, t ≥ 0 � =⇒ � Wt, t ≥ 0 � where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process such that W0 = 0 and with covariance E � WsW T t � = a(β + 1)(1 − a) + aβ (2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a)s � t s �a−β(1−a) 1 dId + β (β(1 − a) − a)(1 − a)s · 1 dId for all 0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We will check that all the conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2] are satisfied, see also [14, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) is satisfied thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4 while the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) directly follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5 and the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) is exactly the statement of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Consequently, we have the convergence in distribution in the Skorokhod space D(R+) such that � VnL⌊nt⌋(u), t ≥ 0 � =⇒ � Wt(u), t ≥ 0 � where (Wt(u))t≥0 is a continuous R2-valued centered Gaussian process such that W0 = 0 and with covariance E � Ws(u)Wt(u)T � = 1 duT uVs for all 0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2), we see that S⌊nt⌋(u) is asymptotically equivalent to N⌊nt⌋(u) + tβ−a(β+1) a(β + 1) β − a(β + 1)(anµn)−1M⌊nt⌋(u) P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Multiplying on the left side by vt = (1, ta(β+1)−β)T , we obtain � 1 √nS⌊nt⌋(u), t ≥ 0 � =⇒ � Wt(u), t ≥ 0 � with Wt(u) = vT t Wt(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hereafter, when 0 ≤ s ≤ t < ∞, we have the covariance E � Ws(u)Wt(u)T � = vT s E � Ws(u)Wt(u)T � vt = 1 d(uT u)vT s Vsvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7) Solving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7), we have E � WsW T t � = 1 dvT s Vsvt for all 0 ≤ s ≤ t < ∞ and the assertion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the almost sure convergence 1 √n log nSn → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We still have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) such that ∥Mn∥2 = o � wn � log wn �1+γ� for all γ > 0 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' However, in the critical regime, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) rather than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3), and wn log n → �Γ(β + 1 + 1 2) Γ(β + 1) �2 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 10 JIAMING CHEN AND LUCILE LAULIN Since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), and since Mn = anYn, we observe for all γ > 0 that ∥Yn∥2 n(log n)2µ2n = o � (log n)−1� log log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In this regard Yn √n log nµn → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8) Similarly, we still have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10) and ∥Nn∥2 = o � n � log n �1+γ� for all γ > 0 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then ∥Nn∥2 n(log n)2 = o � (log n)γ−1� for all γ ∈ (0, 1) P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' and therefore Nn √n log n → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2), we can hereafter conclude that Sn √n log n + a(β + 1) β − a(β + 1) · Yn √n log nµn → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8), the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the quadratic strong law 1 log log n n � k=1 SkST k (k log k)2 → (2β + 1)2 · 1 dId as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We will check that all the conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3] are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) is satisfied thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8 while the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) directly follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9 and the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) is exactly the statement of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Therefore, 1 log � det W −1 n �2 n � k=1 �(det Wk)2 − (det Wk+1)2 (det Wk)2 � WkLk(u)Lk(u)T W T k → 1 duT uW (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9) as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' On the one hand, we have from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='34) 1 log log n n � k=1 �(det Wk)2 − (det Wk+1)2 (det Wk)2 � WkLk(u)Lk(u)T W T k → 1 duT uW as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='33), we have n log n �(det Wk)2 − (det Wk+1)2 (det Wk)2 � → (2β + 1)2 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, we obtain from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9) that 1 log log n n � k=1 uT SkST k u (k log k)2 = 1 log log n n � k=1 wT WkLk(u)Lk(u)T W T k w k log k → (2β + 1)2 d uT u (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10) as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since u ∈ Rd is arbitrary, the assertion follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The MARW admits a scaling limit at the critical regime, or convergence in dis- tribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that � 1 � nt log n S⌊nt⌋, t ≥ 0 � =⇒ � (2β + 1)Bt, t ≥ 0 � MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 11 where (Bt)t≥0 is a continuous d-dimensional canonical Brownian motion with covariance E � BsBT t � = s · 1 dId for all 0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We will check that all the three conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2] are satisfied, see also [42, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' First of all, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7) we know that w−1/2 n ⟨M(u)⟩⌊nt⌋w−1/2 n → t d · uT u as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11) Hence the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Notice that ⌊nt⌋ � k=1 1 wn E � ∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1 � ≤ ⌊nt⌋ � k=1 �w⌊nt⌋ wn �2 1 ϵ2w2 ⌊nt⌋ E � ∆Mk(u)4|Fk−1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='12) since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='21), we observe that ⌊nt⌋ � k=1 ��∆Mk(u)4�� ≤ C1(β)∥u∥4 ⌊nt⌋ � k=1 (akµk)4 ≤ C2(β)∥u∥4 ⌊nt⌋ � k=1 1 k2 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='13) with constants C1(β), C2(β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Therefore, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='13), we have ⌊nt⌋ � k=1 1 wn E � ∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1 � ≤ C3(β)∥u∥4 · t2 ϵ2 · 1 nt(log nt)2 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Simplifying the above expression, we obtain ⌊nt⌋ � k=1 1 wn E � ∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1 � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14) Then the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2), or the Lindeberg condition, is satisfied by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In this particular case at critical regime, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11) implies that the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence � 1 √wn M⌊nt⌋(u), t ≥ 0 � =⇒ � Bt(u), t ≥ 0 � where (Bt(u))t≥0 is a continuous real-valued centered Gaussian process such that B0(u) = 0 and with covariance E � Bs(u)Bt(u) � = s d · uT u for all 0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the critical regime, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) we can write S⌊nt⌋(u) = N⌊nt⌋(u) + (2β + 1) M⌊nt⌋(u) a⌊nt⌋µ⌊nt⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15) From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8) we know that ⟨N(u)⟩⌊nt⌋ nt log n → 0 and N⌊nt⌋(u) � nt log n → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16) Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) again, we conclude that � 1 � nt log n S⌊nt⌋(u), t ≥ 0 � =⇒ � (2β + 1)Bt(u), t ≥ 0 � with E � Bs(u)Bt(u) � = s · uT u d for all 0 ≤ s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='17) Solving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='17), we get E � BsBT t � = s · 1 dId for all 0 ≤ s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 12 JIAMING CHEN AND LUCILE LAULIN which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The superdiffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the almost sure convergence 1 na(β+1)−β Sn → Lβ as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' where the limiting Lβ is an Rd-valued random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In fact, from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8 below, we will see the random vector Lβ is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7), in the superdiffusive regime, we have Tr⟨M⟩n ≤ wn ≤ ∞ � k=1 �Γ(a(β + 1) + 1)Γ(β + k) Γ(a(β + 1) + k)Γ(β + 1) �2 < ∞ for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15], this leads to Mn → M as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' with M = ∞ � k=1 akϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1), Mn = anYn, and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), we observe that Yn na(β+1) → 1 Γ(a(β + 1) + 1)M as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='18) Moreover, equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) still holds and, as 2a(β + 1) > 2β + 1 in the superdiffusive regime, we find that 1 n2(a(β+1)−β) ����Sn + a(β + 1) (β − a(β + 1))µn+1 Yn ���� 2 = o � n−(1−2a(β+1)+2β)� log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), we obtain Sn na(β+1)−β + a(β + 1) β − a(β + 1) · Γ(β + 1)Yn na(β+1) → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='19) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='18), it yields Sn na(β+1)−β → Lβ as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' where Lβ = a(β + 1) a(β + 1) − β · Γ(β + 1) Γ(a(β + 1) + 1)M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='20) and the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the following mean square convergence E ����� 1 na(β+1)−β Sn − Lβ ���� 2� → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='21) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For each test vector u ∈ Rd, we have E � Mn(u)2� = E � ⟨M(u)⟩n � ≤ 1 dwnuT u for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), we obtain sup n≥1 E � Mn(u)2� < ∞ which implies that (Mn(u))n∈N is a martingale bounded in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Therefore E � |Mn(u) − M(u)|2� → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22) MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 13 Moreover, on the one hand (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22) together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='18) implies that E ����� 1 na(β+1) Yn(u) − Y (u) ���� 2� → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='23) On the other hand, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8) we know that E � Nn(u)2� = E � ⟨N(u)⟩n � ≤ 1 d � β β − a(β + 1) �2 nuT u for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since a(β + 1) > β + 1 2 in the superdiffusive regime, we have E ����� 1 na(β+1)−β Nn(u) ���� 2� → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='24) The proof is complete by combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='23) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The expected value of Lβ is E � Lβ � = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='25) whereas its quadratic deviation is E � LβLT β � = � a(β + 1) β − a(β + 1) �2 Γ(β + 1)2Γ(2(a − 1)(β + 1) + 1) Γ((2a − 1)(β + 1) + 1)2 1 dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='26) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The MARW admits a scaling limit at the superdiffusive regime, or convergence in distribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that � 1 na(β+1)−β S⌊nt⌋, t ≥ 0 � =⇒ � Qt, t ≥ 0 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='27) with the limiting Qt = ta(β+1)−βLβ for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all t ≥ 0 and from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='19), we observe that S⌊nt⌋ ⌊nt⌋a(β+1)−β + a(β + 1) β − a(β + 1) · Y⌊nt⌋ ⌊nt⌋a(β+1) → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' which implies 1 na(β+1)−β Sn → ta(β+1)−βLβ as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='28) The P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' convergence in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='28)holds in all finite-dimensional distributions which characterizes the Skorokhod space topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='27) and the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Scaling limit of the barycenter process The study of the scaling limit of the MARW (Sn)n∈N gives us some information on its asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Nonetheless, to understand its pathwise geometric features, we need to discuss its barycenter, or center of mass process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Such topics have been raised and discussed in [36, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In this Section, we turn our attention to the above-mentioned barycenter process (Gn)n∈N defined by Gn := 1 n n � k=1 Sk (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) Our work contains the discussion on the scaling limit and the almost sure convergence in the diffusive, critical and superdiffusive regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The quadratic strong law in the diffusive and crit- ical regimes is also discussed while the mean square convergence in the superdiffusive regime is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 14 JIAMING CHEN AND LUCILE LAULIN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Almost sure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The barycenter process was discussed in [8] for the elephant random walk in dimension d, which is a special case of the process we study here when β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We first begin with the almost sure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the almost sure convergence, in the diffusive regime, 1 nGn → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) while in the critical regime, 1 √n log nGn → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) and, in the superdiffusive regime, 1 na(β+1)−β Gn → 1 1 + a(β + 1) − β Lβ as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) where Lβ was characterized in Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the diffusive regime, from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) we observe that 1 nGn = n � k=1 k n2 · 1 k Sk = n � k=1 1 k Ska′ n,k with a′ n,k = k n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since �n k=1 a′ n,k ≤ 1 for all n ∈ N and the almost sure convergence in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1, from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='12 we can conclude that 1 nGn = n � k=1 1 k Ska′ n,k → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the critical regime, we have from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) that 1 √n log nGn = 1 n3/2 log n n � k=1 Sk = n � k=1 1 √ k log k Ska′′ n,k with a′′ n,k = k1/2 log k n3/2 log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since �n k=1 a′′ n,k ≤ 1 for all n ∈ N and the almost sure convergence in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4 holds, we get from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='12 hat 1 √n log nGn = n � k=1 1 √ k log k Ska′′ n,k → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' and we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Finally, in the superdiffusive regime, we also get from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) that 1 na(β+1)−β Gn = 1 n1+a(β+1)−β n � k=1 Sn = n � k=1 1 ka(β+1)−β Ska′′′ n,k with a′′′ n,k = ka(β+1)−β n1+a(β+1)−β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since n � k=1 a′′′ n,k → 1 1 + a(β + 1) − β as n → ∞ by a simple calculation, and because of the almost sure convergence in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7, we can conclude using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='13 1 na(β+1)−β Gn → 1 1 + a(β + 1) − β Lβ as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Quadratic strong law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 15 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the diffusive regime, we have the quadratic strong law 1 log n n � k=1 GkGT k k2 → 4I(a, β) · 1 dId as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' where I(a, β) is given explicitly I(a, β) = 1 Γ(a(β + 1) + 1)2Γ(β + 1)2 · 2a2(1 − a)(β + 1)3 3(β − a(β + 1))2(1 − a(β + 1) + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We will check that all the three conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2] are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Looking back to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1), we observe that Gn = 1 n n � k=1 Nk − 1 n a(β + 1) β − a(β + 1) n � k=1 1 akµk Mk = 1 n n � k=1 Nk − 1 n a(β + 1) β − a(β + 1) n � k=1 1 akµk k � l=1 alϵl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, changing the order of summation, we have Gn = 1 n n � k=1 Nk − 1 n a(β + 1) β − a(β + 1) n � k=1 akϵk n � l=k 1 alϵl = 1 n n � k=1 Nk − 1 n a(β + 1) β − a(β + 1) n � k=1 ak(δn − δk−1)ϵk where we define δn = �n k=1(akµk)−1 for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moreover, we denote Zn = n � k=1 Nk − a(β + 1) β − a(β + 1) n � k=1 akδk−1ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' such that we have Gn = 1 nZn − δn n · a(β + 1) β − a(β + 1) n � k=1 akϵk = 1 n � Zn − a(β + 1) β − a(β + 1)δnMn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For a fixed text vector u ∈ Rd, we define Hn(u) = � Zn(u) Mn(u) � for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) which implies ∆Hn(u) = Hn+1(u) − Hn(u) = � Nn+1(u)ϵn+1(u)−1 − a(β+1) β−a(β+1)an+1δn an+1 � ϵn+1(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, let Vn = 1 n3/2 � 1 0 0 a(β+1) β−a(β+1)δn � and v = � 1 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then it is immediate that vT VnHn(u) = 1 √nGn for all n ∈ N (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) and that lim n→∞ Vn⟨H(u)⟩nV T n = lim n→∞ 1 n3 � 1 −1 −1 1 � n−1 � k=1 � a(β + 1) β − a(β + 1) �2 δ2 ka2 k+1E � ϵk+1(u)2|Fk � = lim n→∞ 1 n3 · a2(1 − a)(β + 1)3uT u d(β − a(β + 1))2(1 − a(β + 1) + β) � 1 −1 −1 1 � n−1 � k=1 δ2 ka2 k+1µ2 k+1 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 16 JIAMING CHEN AND LUCILE LAULIN By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), we know that n−(1+a(β+1)−β)δn → 1 1 + a(β + 1) − β · 1 Γ(a(β + 1) + 1)Γ(β + 1) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence the above calculation leads us to Vn⟨H(u)⟩nV T n → I(a, β)uT u · 1 d � 1 −1 −1 1 � as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7) where I(a, β) = 1 1 − 2(a(β + 1) − β) · a2(1 − a)(β + 1)3 (β − a(β + 1))2(1 − a(β + 1) + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8) Consequently, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7) ensures that the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Notice that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1), there exists some constant C1(a, β) > 0 and similarly, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22), there exists some other constant C2(a, β) > 0 such that ∥Nn∥2 ≤ C1(a, β)n2 and a2 kϵk(u)2 ≤ C2(a, β)n2δ−2 n for all 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moreover, notice that for all 1 ≤ k ≤ n, Vn∆Hk(u) = 1 n3/2 � Nk(u)ϵk(u)−1 − a(β+1) β−a(β+1)akδk−1 a(β+1) β−a(β+1)akδn � ϵk(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, for all 1 ≤ k ≤ n, we observe that ∥Vn∆Hk(u)∥2 ≤ 4a2 k n3 � a(β + 1) β − a(β + 1) �2��β − a(β + 1) aka(β + 1) Nk(u) ϵk(u) �2 + δ2 k−1 + δ2 n � ϵk(u)2 ≤ C(a, β) n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9) for some constant C(a, β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Consequently, we n � k=1 E � ∥Vn∆Hk(u)∥4� ≤ 1 nC(a, β) → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' since, for all ϵ > 0, n � k=1 E � ∥Vn∆Hk(u)∥21{∥Vn∆Hk(u)∥>ϵ}|Fk−1 � ≤ 1 ϵ2 n � k=1 E � ∥Vn∆Hk(u)∥4� → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10) Then the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2), or the Lindeberg condition, is satisfied by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hereafter, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), and by the definition of δn, we know there exists some constant C′(a, β) ̸= 0 such that log � det V −1 n �2 log n → C′(a, β) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This ensures that there exists some other constant C′′(a, β) > 0 such that ∞ � n=1 1 � log � det V −1 n �2�2 E �� ∥Vn∆Hn(u)∥4|Fn−1 � ≤ C2(a, β) ∞ � n=1 1 (log n)2 E � ∥Vn∆Hn(u)∥4|Fn−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Finally, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9) leads to ∞ � n=1 1 (log n)2 ∥Vn∆Hn(u)∥4 ≤ C(a, β) ∞ � n=1 1 (n log n)2 < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 17 for some constant C(a, β) > 0 depending only on a and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) is satisfied by combining the above with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' On the one hand, 1 log � det V −1 n �2 n � k=1 �(det Vk)2 − (det Vk+1)2 (det Vk)2 � VkHk(u)Hk(u)T V T k → 1 duT uV (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11) as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' where V = � 1 −1 −1 1 � I(a, β) and I(a, β) has been specified in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, we have 1 log n n � k=1 �(det Vk)2 − (det Vk+1)2 (det Vk)2 � VkHk(u)Hk(u)T V T k → 4 − 2(a(β + 1) − β) d uT uV as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' since log n log � det V −1 n �2 → 4 − 2(a(β + 1) − β) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), we have n �(det Vn)2 − (det Vn+1)2 (det Vn)2 � → 4 − 2 � a(β + 1) − β � as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11), we observe that 1 log n n � k=1 uT GkGT k u k2 = 1 log n n � k=1 vT VkHk(u)Hk(u)T V T k v k → vT V v · 1 duT u as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since u ∈ Rd is arbitrary, the assertion follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the critical regime, we have the quadratic strong law 1 log log n n � k=1 GkGT k (k log k)2 → 4(2β + 1)2 9 1 dId as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We will check that all the three conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2] are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Denote Wn = 1 n√n log n � 1 0 0 a(β+1) β−a(β+1)δn � and w = � 1 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, for H defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), it is clear that wT WnHn(u) = 1 √n log nGn for all n ∈ N and that lim n→∞ Wn⟨H(u)⟩nW T n = lim n→∞ 1 n3 log n � 1 −1 −1 1 � n−1 � k=1 (2β + 1)2δ2 ka2 k+1E � ϵk+1(u)2|Fk � = lim n→∞ (2β + 1)2 n3 log n · uT u d � 1 −1 −1 1 � n−1 � k=1 δ2 ka2 k+1µ2 k+1 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), we know that n−3/2δn → 2 3 · Γ(β + 1) Γ(β + 1 + 1 2) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 18 JIAMING CHEN AND LUCILE LAULIN Hence, the above calculation leads us to Wn⟨H(u)⟩nW T n → I(β)uT u · 1 d � 1 −1 −1 1 � as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' with I(β) = 4(2β + 1)2 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='12) Consequently, the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) is satisfied thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Notice that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1), there exists some constant C1(β) > 0 and similarly, there exists some constant C2(β) > 0 such that ∥Nn∥2 ≤ C1(β)n2 and a2 kϵk(u)2 ≤ C2(β)n2δ−2 n log n for all 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, notice for all 1 ≤ k ≤ n that Wn∆Hk(u) = 1 n√n log n � Nk(u)ϵk(u)−1 − a(β+1) β−a(β+1)akδk−1 a(β+1) β−a(β+1)akδn � ϵk(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The ensures that, for all 1 ≤ k ≤ n, ∥Wn∆Hk(u)∥2 ≤ 4a2 k n3 log n(2β + 1)2 �� (2β + 1)−2 Nk(u) ϵk(u) �2 + δ2 k−1 + δ2 n � ϵk(u)2 ≤ C(β) n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='13) for some constant C(β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, n � k=1 E � ∥Wn∆Hk(u)∥4� ≤ 1 nC(β) → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' since, for all ϵ > 0, n � k=1 E � ∥Wn∆Hk(u)∥21{∥Wn∆Hk(u)∥>ϵ}|Fk−1 � ≤ 1 ϵ2 n � k=1 E � ∥Wn∆Hk(u)∥4� → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14) Therefore, the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2), or the Lindeberg condition, is satisfied using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hereafter, we know that log � det W −1 n �2 log log n → 4 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This ensures that there exists some constant C2(β) > 0 such that ∞ � n=1 1 � log � det W −1 n �2�2 E � ∥Wn∆Hn(u)∥4|Fn−1 � ≤ ∞ � n=1 C2(β) (log log n)2 E � ∥Wn∆Hn(u)∥4|Fn−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15) We get from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='13) that ∞ � n=1 1 (log log n)2 ∥Wn∆Hn(u)∥4 ≤ C(β) ∞ � n=1 1 (n log n log log n)2 < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' for some constant C(β) > 0 depending only onβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) is satisfied using the above together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, 1 log � det W −1 n �2 n � k=1 �(det Wk)2 − (det Wk+1)2 (det Wk)2 � WkHk(u)Hk(u)T W T k → 1 duT uW as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' where W = 4(2β + 1)2 9 � 1 −1 −1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 19 Furthermore, on the one hand we have 1 log log n n � k=1 �(det Wk)2 − (det Wk+1)2 (det Wk)2 � WkHk(u)Hk(u)T W T k → 1 duT uW as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' since log log n log � det W −1 n �2 → 1 4 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' On the other hand, we have n log n �(det Wn)2 − (det Wn+1)2 (det Wn)2 � → 1 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11), we observe that 1 log log n n � k=1 uT GkGT k u (k log k)2 = 1 log log n n � k=1 wT WkHk(u)Hk(u)T W T k w 4k log k → wT Ww · 1 4duT u (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16) as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since u ∈ Rd is arbitrary, the assertion follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the superdiffusive regime, we have the mean square convergence, given by E ����� 1 na(β+1)−β Gn − 1 1 + a(β + 1) − β Lβ ���� 2� → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all test vector u ∈ Rd, it is immediate that E ����� 1 na(β+1)−β Gn(u) − 1 1 + a(β + 1) − β Lβ(u) ���� 2� ≤ 2E ����� 1 n1+a(β+1)−β Zn(u) ���� 2� + 2E ����� 1 n1+a(β+1)−β · a(β + 1) a(β + 1) − β δnMn − 1 1 + a(β + 1) − β Lβ ���� 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='18) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='20) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7), the second term converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Looking back to the first term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='18), we observe E ����� 1 n1+a(β+1)−β Zn(u) ���� 2� ≤ 4 n1+2(a(β+1)−β) n � k=1 E � Nk(u)2� + 4 n1+2(a(β+1)−β) � a(β + 1) a(β + 1) − β �2 E ������ n � k=1 akδk−1ϵk(u) ����� 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='19) The first term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='19) converges to zero because E[Nk(u)] ≤ (uT u)n for all 1 ≤ k ≤ n, and moreover, in the superdiffusive regime we have a(β +1) > β +1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The second term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='19) also converges to zero because n−(1+a(β+1)−β)δn → 1 1 + a(β + 1) − β · 1 Γ(1 + a(β + 1))Γ(β + 1) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Finally, using the above and that M(u) = �∞ k=1 akϵk(u) is bounded in L2, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The barycenter process admits a scaling limit at the diffusive regime, or conver- gence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that � 1 √nG⌊nt⌋, t ≥ 0 � =⇒ � 1 � 0 Wtr dr, t ≥ 0 � 20 JIAMING CHEN AND LUCILE LAULIN where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process define in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3 with its covariance defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In particular, E �� 1 � 0 Wsv dv �� 1 � 0 Wtu du �T � = β 3(β(1 − a) − a)(1 − a)s · 1 dId + 2(a(β + 1)(1 − a) + aβ) 3(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a)(1 + (1 − a)(β + 1))ta−β(1−a)s1−a+β(1−a) · 1 dId (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='20) for all 0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' An easy calculation leads to lim n→∞ 1 √nG⌊nt⌋ = lim n→∞ 1 � 0 1 √nS⌊ntr⌋ dr =⇒ 1 � 0 Wtr dr which ensures that G⌊nt⌋ is a continuous function of S⌊ntr⌋ in D([0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, the last convergence in law is due to the functional central limit Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3, with (Wt)t≥0 defined there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, the barycenter process (Gn)n∈N admits a Gaussian scaling limit in the diffusive regime as well, with covariance E �� 1 � 0 Wsv dv �� 1 � 0 Wtu du �T � = 2 1 � 0 u � 0 E � WsvW T tu � dv du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='20) and the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The barycenter process admits a scaling limit at the critical regime, or convergence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that � 1 � nt log n G⌊nt⌋, t ≥ 0 � =⇒ � 1 � 0 (2β + 1)Btr dr, t ≥ 0 � where (Bt)t≥0 is a continuous Rd-valued centered Gaussian process define in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6 with its covariance defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For each r ∈ [0, 1], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11) implies that lim n→∞ 1 � nt log n M⌊ntr⌋(u) a⌊ntr⌋µ⌊ntr⌋ = lim n→∞ 1 � nt log n � ntr(log n + r t log r) �1/2Btr(u) P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' for all u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moreover, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16) yields lim n→∞ 1 � nt log n N⌊ntr⌋(u) = lim n→∞ r1/2 · 1 � ntr log n N⌊ntr⌋(u) = 0 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' for all u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15), we have � 1 � nt log n S⌊ntr⌋(u), t ≥ 0 � =⇒ � (2β + 1)Btr(u), t ≥ 0 � for all u ∈ Rd and r ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, we use again lim n→∞ 1 � nt log n G⌊nt⌋ = lim n→∞ 1 � 0 1 � nt log n S⌊ntr⌋ dr =⇒ 1 � 0 (2β + 1)Btr dr and the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 21 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The barycenter process admits a scaling limit at the superdiffusive regime, or convergence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that � 1 na(β+1)−β G⌊nt⌋, t ≥ 0 � =⇒ � 1 � 0 Qtr dr, t ≥ 0 � with the covariance specified in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) and the limiting Lβ characterized in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8 and Qt = ta(β+1)−βLβ characterized in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Again, we find that lim n→∞ 1 na(β+1)−β G⌊nt⌋ = 1 � 0 1 na(β+1)−β S⌊ntr⌋ dr =⇒ 1 � 0 Qtr dr which ensures that G⌊nt⌋ is a continuous function of S⌊ntr⌋ in D([0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, the last convergence in law is due to the functional central limit Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence the barycenter process (Gn)n∈N admits a non-degenerate scaling limit in the superdiffusive regime as well, with covariance E �� 1 � 0 Qsv dv �� 1 � 0 Qtu du �T � = 2 1 � 0 u � 0 E � QsvQT tu � dv du = ta(β+1)−βsa(β+1)−β (1 + a(β + 1) − β)2 E � LβLT β � = ta(β+1)−βsa(β+1)−β (1 + a(β + 1) − β)2 � a(β + 1) β − a(β + 1) �2 Γ(2(a − 1)(β + 1) + 1) Γ((2a − 1)(β + 1) + 1)2 · 1 dId for all 0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Velocity of quadratic mean displacement In this Section, we investigate the velocity of the mean square displacement of the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This quantitative estimates give us the information on how fast the limit Theorems in Section 4 are carried on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Similar convergence velocities have been discussed in [20, 26], where the authors analyzed the convergence velocity of the moments of a one-dimensional elephant random walk of all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the superdiffusive regime, the convergence velocity was discussed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Here, only the rate of quadratic moment convergence for the MARW in all of the three (diffusive, critical, and superdiffusive) regimes are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Following the limit Theorems in Section 4, we expect the asymptotic behavior of the mean square displacement is as follows, E � SnST n � ∼ � � � � � � � � � � � n · (a−2β)(1−a)(β+1)+β(a+1) (2(β+1)(1−a)−1)(a−β(1−a))(1−a) · 1 dId when a < 1 − 1 2(β+1) n log n · (2β + 1)2 · 1 dId when a = 1 − 1 2(β+1) n2(a(β+1)−β) · � a(β+1) β−a(β+1) �2 Γ(2(a−1)(β+1)+1) Γ((2a−1)(β+1)+1)2 · 1 dId when a > 1 − 1 2(β+1), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) where the notation ∼ indicates two sequences an ∼ bn if and only if an/bn → 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The aim of this Section is not only to show that the above estimates (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) are valid, but also to investigate the exact velocity of their convergence in the diffusive and critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all p < (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞, 1 nE � SnST n � − (a − 2β)(1 − a)(β + 1) + β(a + 1) (2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a) · 1 dId ∼ −(C1n−2(1−a)(β+1) + C2n−1) · 1 dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 22 JIAMING CHEN AND LUCILE LAULIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Take the vector v = (1, −1)T and Vn ∈ R2×2 as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, 1 √nSn(u) = vT VnLn(u), where Ln(u) = (Nn(u), Mn(u))T is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In particular, 1 nuT E � SnST n � u = vT VnE � Ln(u)Ln(u)T � V T n v = vT VnE � � E � Nn(u)2� E � Nn(u)Mn(u) � E � Mn(u)Nn(u) � E � Mn(u)2� � � V T n v = vT VnE � � E � ⟨N(u)⟩n � E � ⟨N(u), M(u)⟩n � E � ⟨M(u), N(u)⟩n � E � ⟨M(u)⟩n � � � V T n v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Therefore, 1 nuT E � SnST n � u = 1 nE � ��N(u)⟩n � + 1 na2nµ2n � a(β + 1) β − a(β + 1) �2 E � ⟨M(u)⟩n � − 2 nanµn � a(β + 1) β − a(β + 1) � E � ⟨M(u), N(u)⟩n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since the test vector u ∈ Rd is taken arbitrarily, we get from Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16 that 1 nE � SnST n � − (a − 2β)(1 − a)(β + 1) + β(a + 1) (2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a) · 1 dId ∼ −(C1n−2(1−a)(β+1) + C2n−1) · 1 dId as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' When p = (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞, 1 n log nE � SnST n � − (2β + 1)2 · 1 dId ∼ −(C1(log n)−1 + C2n−1) · 1 dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Take w = (1, −1)T and Wn ∈ R2×2 as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then 1 √n log nSn(u) = wT WnLn(u) as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='29) for all u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In particular, 1 n log nuT E � SnST n � u = wT WnE � Ln(u)Ln(u)T � W T n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, 1 n log nuT E � SnST n � u = wT WnE � � E � ⟨N(u)⟩n � E � ⟨N(u), M(u)⟩n � E � ⟨M(u), N(u)⟩n � E � ⟨M(u)⟩n � � � W T n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Therefore, we get by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) as n → ∞, 1 n log nuT E � SnST n � u = 1 n log n � E � ⟨N(u)⟩n � + (2β + 1)2 a2nµ2n E � ⟨M(u)⟩n �� , which implies 1 n log nE � SnST n � − (2β + 1)2 · 1 dId ∼ −(C1(log n)−1 + C2n−1) · 1 dId as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Superdiffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 23 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' When p > (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞, 1 n2(a(β+1)−β) E � SnST n � − � a(β + 1) β − a(β + 1) �2 Γ(2(a − 1)(β + 1) + 1) Γ((2a − 1)(β + 1) + 1)2 · 1 dId ∼ −(C1n−4(a(β+1)−β)+1 + C2n−2(a(β+1)−β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Similar to previous computations for the diffusive regime, we have for all u ∈ Rd, 1 n2(a(β+1)−β) uT E � SnST n � u = 1 n2(a(β+1)−β) E � ⟨N(u)⟩n � + 1 n2(a(β+1)−β)a2nµ2n � a(β + 1) β − a(β + 1) �2 E � ⟨M(u)⟩n � − 2 n2(a(β+1)−β)anµn � a(β + 1) β − a(β + 1) � E � ⟨M(u), N(u)⟩n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and since u ∈ Rd is arbitrary, 1 n2(a(β+1)−β) E � SnST n � − � a(β + 1) β − a(β + 1) �2 Γ(2(a − 1)(β + 1) + 1) Γ((2a − 1)(β + 1) + 1)2 · 1 dId ∼ −(C1n−4(a(β+1)−β)+1 + C2n−2(a(β+1)−β)) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Cram´er moderate deviations In this Section, we discuss the Cram´er moderate deviations for the multidimensional reinforced random walk (Sn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The similar statistical quantity as well as the Berry-Esseen bound for the one-dimensional elephant random walk (ERW) without amnesia-reinforcement has been given in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Our derivation of Cram´er moderate deviations for the MARW does not rely on a Berry- Esseen bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The discussion of such statistical quantities is expected to reveal the transience property and the central limit Theorems for the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For this direction, readers are refereed to [3, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='21 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22, we can properly state the Cram´er moderate deviations principles for the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the diffusive and critical regimes, we have the following Cram´er moderate deviations for the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Let (ϑn)n∈N ⊆ R be a non-decreasing sequence so that ϑn/√n → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Take any non-empty Borel set B ⊆ Rd, then we have − inf x∈int B 1 2∥x∥2 ≤ lim inf n→∞ ϑ−2 n log P �anµnSn ϑn√wn ∈ B � ≤ lim sup n→∞ ϑ−2 n log P �anµnSn ϑn√wn ∈ B � ≤ − inf x∈cl B 1 2∥x∥2, where int B and cl B denote the interior and the closure of B ⊆ Rd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Our proof will only present the Cram´er moderate deviations for the MARW in the diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The same property for the critical regime follows from exactly the same steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' First, take xB = infx∈B ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then it is obvious that infx∈cl B ∥x∥ ≤ xB and infx∈cl B ∥x∥2/2 ≤ x2 B/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Henceforth, P �anµnSn ϑn√wn ∈ B � ≤ d � j=1 P ����� anµnSj n √wn ���� ≥ ϑnxB d � ≤ � 1 − Φ(ϑnxB) � F(B, ϑ, n), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) 24 JIAMING CHEN AND LUCILE LAULIN where we write F(B, ϑ, n) := 2Cd · exp � 1 √n � ϑnxB 2d �3 + 1 n � ϑnxB 2d �2 + 1 √n(1 + 1 2 log n)(1 + ϑnxB 2d ) � + 2Cd · exp � 1 √n � ϑnxB 2d �3 + 1 n2(1−a)(β+1) � ϑnxB 2d �2 + 1 √n(1 + 1 2 log n)(n1/2−(1−a)(β+1) + ϑnxB 2d ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, lim sup n→∞ ϑ−2 n log P �anµnSn ϑn√wn ∈ B � ≤ −1 2x2 B ≤ − inf x∈cl B 1 2∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' To achieve the asymptotic lower bound, we first notice that this assertion automatically holds if int B = ∅, whence − infx∈∅ ∥x∥2/2 = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Consequently, we assume that int B ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Notice that int B is open in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, for all ϵ∗ > 0 sufficiently small, we could find x∗ ∈ int B with 0 < 1 2∥x∗∥2 < inf x∈int B 1 2∥x∥2 + ϵ∗ and 0 < min ���xj ∗ �� : 1 ≤ j ≤ d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Choose ϵ∗∗ sufficient small such that 0 < ϵ∗∗ < ���xj ∗ ��� for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, U(x∗, ϵ∗∗) ⊆ int B ⊆ B, where U(x∗, ϵ∗∗) := � x ∈ Rd : ��xj − xj ∗ �� < ϵ∗∗ for all j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' On the other hand, P �anµnSn ϑn√wn ∈ B � ≥ P �anµnSn √wn ∈ ϑn · U(x∗, ϵ∗∗) � ≥ d � j=1 P � ϑn(xj ∗ + ϵ∗∗) ≥ anµnSj n √wn ≥ ϑn(xj ∗ − ϵ∗∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='21 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22, we know that lim n→∞ P �anµnSj n √wn ≥ ϑn(xj ∗ + ϵ∗∗) �� P �anµnSj n √wn ≥ ϑn(xj ∗ − ϵ∗∗) � = 0 for each j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Similar to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1), lim inf n→∞ ϑ−2 n log P �anµnSn ϑn√wn ∈ B � ≥ −1 2∥x∗ − ϵ∗∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Letting ϵ∗∗ → 0, we observe that lim inf n→∞ ϑ−2 n log P �anµnSn ϑn√wn ∈ B � ≥ −1 2∥x∗∥2 ≥ − inf x∈int B 1 2∥x∥2 − ϵ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since ϵ∗ > 0 was take arbitrarily, letting ϵ∗ → 0, we verify the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Technical Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Asymptotics of the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We start by introducing the following processes that are of great influence on the behavior of the random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Let (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , ed) denote a canonical Euclidean basis of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For each n ∈ N and 1 ≤ j ≤ d, define N X n (j) = n � k=1 1{Xj k̸=0}µk and Σn = d � j=1 N X n (j)ejeT j , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) such that (Σn)n∈N is a matrix-valued process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the following almost sure convergence in the three regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 1 nµn+1 Σn → 1 d(β + 1)Id as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2) MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For each n ∈ N and 1 ≤ j ≤ d, define ΛX n (j) = N X n (j) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) It follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) that ΛX n+1(j) = n n + 1ΛX n (j) + 1 n + 11{Xj n+1̸=0}µn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moreover, we observe thanks to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='12) that ΛX n+1(j) = n n + 1 · γnΛX n (j) + 1 n + 11{Xj n+1̸=0}µn+1 − a(β + 1) n + 1 ΛX n (j) = n n + 1 · γnΛX n (j) + µn+1 n δX n+1(j) + (1 − a)µn+1 d(n + 1) with δX n+1(j) = 1{Xj n+1̸=0} − P � Xj n+1 ̸= 0|Fn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) we know ΛX n (j) = 1 nan � ΛX 1 (j) + 1 − a d n � k=2 akµk + HX n (j) � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4) with HX n (j) = n � k=2 akµkδX k (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It is clear that for a fixed 1 ≤ j ≤ d, the real-valued process (HX n (j))n∈N is locally square-integrable since it is a finite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Afterwards, this process appears to be a martingale adapted to (Fn)n∈N because (δX n (j))n∈N satisfied the martingale difference relation E[δX n+1(j)|Fn] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It is obvious that ⟨HX(j)⟩n ≤ wn = n � k=1 (akµk)2 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, we get by [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15] that for all γ > 0 HX n (j)2 ⟨HX(j)⟩n = o �� log⟨HX(j)⟩n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) Since ⟨HX(j)⟩n ≤ wn and by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5), we obtain that HX n (j)2 = o � wn � log wn �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the diffusive regime, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3), we have HX n (j)2 = o � n1−2(a(β+1)−β)� log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), we observe that � HX n (j) nanµn+1 �2 = o � n−1� log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence HX n (j) nanµn+1 → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) again, we observe further 1 nanµn+1 n � k=1 akµk → 1 (1 − a)(β + 1) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) 26 JIAMING CHEN AND LUCILE LAULIN Hence, we have µ−1 n+1ΛX n (j) →→ 1 β + 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4), we can then conclude that 1 nµn+1 Σn → 1 d(β + 1)Id as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' in the diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the critical regime, where a = 1 − 1 2(β+1), we have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4)) HX n (j)2 = o � log n � log log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence � HX n (j) nanµn+1 �2 = o � n−1 log n � log log n �1+γ� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' which implies that HX n (j) nanµn+1 → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Similar to the convergence in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), in the critical regime, we observe 1 nanµn+1 n � k=1 akµk → 1 2 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, we conclude that µ−1 n+1ΛX n (j) → 1 d(β + 1) and 1 nµn+1 Σn → 1 d(β + 1)Id as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' which proves (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' In the superdiffusive regime, we have HX n (j)2 = o � 1 � P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' and then � HX n (j) nanµn+1 �2 = o � n−2(1−a)(β+1)� P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' which implies HX n (j) nanµn+1 → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We can similarly show that µ−1 n+1ΛX n (j) → 1 β + 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' which then ensures that 1 nµn+1 Σn → 1 d(β + 1)Id as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Consequently, the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ The next result follows directly from the definition of Mn and Nn Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the following formulas for the predictable matrix-valued quadratic varia- tions ⟨M⟩n = (a1µ1)2E � X1XT 1 � + n−1 � k=1 a(β + 1) ka−2 k+1 µk+1Σk + 1 − a da−2 k+1 µ2 k+1Id − �γk − 1 a−1 k+1 �2 YkY T k , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7) and ⟨N⟩n = � β β − a(β + 1) �2 E � X1XT 1 � + n−1 � k=1 a(β + 1) kµk+1 Σk + 1 − a d Id − �γk − 1 µk+1 �2 YkY T k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8) MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 27 In particular, we have Tr⟨M⟩n = wn − n � k=1 (γk − 1)2a2 k+1∥Yk∥2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9) and Tr⟨N⟩n = � β β − a(β + 1) �2 n − n−1 � k=1 �a(β + 1) kµk+1 �2 ∥Yk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10) Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We have the following estimate for the matrix-valued conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' E � ϵn+1ϵT n+1|Fn � = a(β + 1) n µn+1Σn + 1 − a d µ2 n+1Id − (γn − 1)2YnY T n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' And as a consequence E � ∥ϵn+1∥2|Fn � = µ2 n+1 − (γn − 1)2∥Yn∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Observe that E � ϵn+1ϵT n+1|Fn � = E � Yn+1Y T n+1|Fn � − γ2 nYnY T n with E � Yn+1Y T n+1|Fn � = YnY T n + 2µn+1YnE � XT n+1|Fn � + µ2 n+1E � Xn+1XT n+1|Fn � = � 1 + 2a(β + 1) n � YnY T n + µ2 n+1E � Xn+1XT n+1|Fn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11) For all k ≥ 1, we know that XkXT k = �d j=1 1{Xj k̸=0}ejeT j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then P � Xj n+1 ̸= 0|Fn � = n � k=1 P � βn+1 = k � P � (AnXk)j ̸= 0|Fn � = n � k=1 1{Xj k̸=0}P � An = ±Id � (β + 1)µk nµn+1 + n � k=1 � 1 − 1{Xj k̸=0} � P � An = ±Jd � (β + 1)µk nµn+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence P � Xj n+1 ̸= 0|Fn � = β + 1 nµn+1 � P � An = +Id � − P � An = +Jd �� N X n (j) + 2P � An = +Jd � = a(β + 1) nµn+1 N X n (j) + 1 − a d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='12) Therefore E � Xn+1XT n+1|Fn � = d � j=1 P � Xj n+1 ̸= 0|Fn � ejeT j = a(β + 1) nµn+1 Σn + 1 − a d Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='13) And from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='13) we can conclude that E � ϵn+1ϵT n+1|Fn � = E � Yn+1Y T n+1|Fn � − γ2 nYnY T n = � 1 + 2a(β + 1) n � YnY T n + a(β + 1) n µn+1Σn + 1 − a d µ2 n+1Id − γ2 nYnY T n = a(β + 1) n µn+1Σn + 1 − a d µ2 n+1Id − (γn − 1)2YnY T n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14) On the other hand Tr(Σn) = nµn+1 β + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15) Taking traces in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14) and by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15), we have E � ∥ϵn+1∥2|Fn � = µ2 n+1 − (γn − 1)2∥Yn∥2 which ensures that the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 28 JIAMING CHEN AND LUCILE LAULIN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Scaling limits of the random walk and the barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For each n ∈ N and test vector u ∈ Rd, let Vn = 1 √n � 1 0 0 a(β+1) β−a(β+1)(anµn)−1 � and v = � 1 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16) Then vT VnLn(u) = 1 √nSn(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='17) And for all t ≥ 0, we have Vn⟨L(u)⟩⌊nt⌋V T n → uT u d Vt as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='18) where Vt = 1 (β − a(β + 1))2 � β2t aβ 1−at1+β−a(β+1) aβ 1−at1+β−a(β+1) a2(β+1)2 1−2a(β+1)+2β t1+2β−2a(β+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3 and the fact that ⟨M(u)⟩n = uT ⟨M⟩nu, we see that ⟨M(u)⟩⌊nt⌋ = a2 1µ2 1uT E � X1XT 1 � u + ⌊nt⌋−1 � k=1 a(β + 1) k a2 k+1µk+1uT Σku + 1 − a d a2 k+1µ2 k+1uT u − (γk − 1)2a2 k+1uT YkY T k u and ⟨N(u)⟩⌊nt⌋ = � β β − a(β + 1) �2 uT E � X1XT 1 � u + � β β �� a(β + 1) �2 ⌊nt⌋−1 � k=1 a(β + 1) kµk+1 uT Σku + 1 − a d uT u − �γk − 1 µk+1 �2 uT YkY T k u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Using a similar token and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1, we can work out the off-diagonal entries in ⟨L(u)⟩⌊nt⌋, and we obtain that lim n→∞ Vn⟨L(u)⟩⌊nt⌋V T n = lim n→∞ uT u nd(β − a(β + 1))2 � � � β2⌊nt⌋ a(β+1)β anµn �⌊nt⌋−1 k=0 ak+1µk+1 a(β+1)β anµn �⌊nt⌋−1 k=0 ak+1µk+1 � a(β+1) anµn �2 �⌊nt⌋−1 k=0 (ak+1µk+1)2 � � � = uT u d(β − a(β + 1))2 � β2t aβ 1−at1−(a(β+1)−β) aβ 1−at1−(a(β+1)−β) a2(β+1)2 1−2(a(β+1)−β)t1−2(a(β+1)−β) � = uT u d Vt P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' where the last equality is due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Thus, it implies that 1 nanµn n � k=1 akµk → 1 1 − (a(β + 1) − β) and 1 n(anµn)2 n � k=1 (akµk)2 → 1 1 − 2(a(β + 1) − β) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='18) holds and the assertion is then verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The MARW satisfies the Lindeberg condition in the diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' That is, for all t ≥ 0 and all ϵ > 0, ⌊nt⌋ � k=1 E � ∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1 � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' On the one hand, it is easy to compute from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16) that, for all 1 ≤ k ≤ n, Vn∆Lk(u) = 1 √n(β − a(β + 1))µn � β µn µk a ak an � ϵk(u) which implies ∥Vn∆Lk(u)∥2 = 1 n(β − a(β + 1))2 �β2 µ2 k + a2a2 k (anµn)2 � ϵk(u)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence ∥Vn∆Lk(u)∥4 ≤ 2 n2(β − a(β + 1))4 �β4 µ4 k + a4a4 k (anµn)4 � ϵk(u)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='20) On the other hand, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) we observe that 1 na2n n � k=1 a2 k ≤ C1(a, β)−1 and 1 na4n n � k=1 a4 k ≤ C2(a, β)−1 for all n ∈ N (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='21) and where C1(a, β), C2(a, β) > 0 are constants depending only on a and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moreover, we get that sup 1≤k≤n |ϵk(u)| ≤ sup 1≤k≤n ∥ϵk∥∥u∥ ≤ sup 1≤k≤n (β + 2)µk∥u∥ ≤ (β + 2)µn∥u∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22) Hence, we deduce from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='21) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22) n � k=1 ∥Vn∆Lk(u)∥4 ≤ 2 n2(β − a(β + 1))4 �� β(β + 2) �4∥u∥4 + � a(β + 2) �4∥u∥4 C2(a, β) � → 0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='23) as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This implies that n � k=1 E � ∥Vn∆Lk(u)∥4|Fk−1 � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Therefore, for all ϵ > 0, we obtain n � k=1 E � ∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1 � ≤ 1 ϵ2 n � k=1 E � ∥Vn∆Lk(u)∥4|Fk−1 � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This yields finally ⌊nt⌋ � k=1 E � ∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1 � ≤ 1 ϵ2 ⌊nt⌋ � k=1 E ����(VnV −1 ⌊nt⌋)V⌊nt⌋∆Lk(u) ��� 4 |Fk−1 � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' since VnV −1 ⌊nt⌋ converges as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The deterministic matrix Vt defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='19) can be rewritten as Vt = tα1K1 + tα2K2 + · · · + tαqKq with q ∈ N, αj > 0 and each Kj is a symmetric matrix for all 1 ≤ j ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' A direct computation analoguous to the one in [32] shows that Vt = tα1K1+tα2K2+tα3K3, where α1 = 1, α2 = 1 − a(β + 1) > 0, α3 = 1 − 2a(β + 1) > 0 since a < 1 − 1 2(β+1) is in the diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moreover K1 = β2 (a(β + 1) − β)2 � 1 0 0 0 � , K2 = aβ (1 − a)(a(β + 1) − β)2 � 0 1 1 0 � , K3 = a2(β + 1)2 (1 − 2a(β + 1) + 2β)(a(β + 1) − β)2 � 0 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 30 JIAMING CHEN AND LUCILE LAULIN □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Given the matrix-valued process (Vn)n∈N define in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16), we have ∞ � n=1 1 � log � det V −1 n �2�2 E � ∥Vn∆Ln(u)∥4|Fn−1 � < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16), it is immediate that det V −1 n = β − a(β + 1) a(β + 1) nanµn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='24) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), we obtain log � det V −1 n �2 log n → 2(1 − a)(β + 1) as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='25) Hence there exists a constant C(a, β) > 0 depending only on a and β such that ∞ � n=1 1 � log � det V −1 n �2�2 E � ∥Vn∆Ln(u)∥4|Fn−1 � ≤ C(a, β) ∞ � n=1 1 (log n)2 E � ∥Vn∆Ln(u)∥4|Fn−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='26) Hereafter, equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='20), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='23) together imply that ∞ � n=1 1 (log n)2 ∥Vn∆Ln(u)∥4 ≤ C′(a, β) ∞ � n=1 1 (n log n)2 < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='27) for some other constant C′(a, β) > 0 depending only on a and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Consequently, equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='27) together (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='26) ensures that the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For each n ∈ N and test vector u ∈ Rd, let Wn = 1 √n log n � 1 0 0 2β+1 anµn � and w = � 1 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='28) Then for all t ≥ 0, we have wT WnLn(u) = 1 √n log nSn(u) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='29) and Wn⟨L(u)⟩nW T n → uT u d W as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' where Wt = (2β + 1)2 � 0 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='30) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' It is clear that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='29) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Using a similar token than for the proof Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4, we have lim n→∞ Wn⟨L(u)⟩nW T n = lim n→∞ 4uT u (n log n)d � � � β2n β(β+ 1 2 ) anµn �n−1 k=0 ak+1µk+1 β(β+ 1 2 ) anµn �n−1 k=0 ak+1µk+1 � β+ 1 2 anµn �2 �n−1 k=0(ak+1µk+1)2 � � � = 4uT u d � 0 0 0 � β + 1 2 �2 � = uT u d W P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 31 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The MARW satisfies the Lindeberg condition in the critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' That is, for all t ≥ 0 and all ϵ > 0, given the (Wn)n∈N defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16), it satisfies n � k=1 E � ∥Wn∆Lk(u)∥21{∥WnLk(u)∥2>ϵ}|Fk−1 � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' We state that equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='20) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='21) remain true with Vn replaced by Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' More precisely, they can be rewritten as ∥Wn∆Lk(u)∥4 ≤ 32 (n log n)2 �β4 µ4 k + a4a4 k (anµn)4 � ϵk(u)4 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='31) and 1 na4n n � k=1 a4 k ≤ C(a, β)−1 for all n ∈ N where C(a, β) > 0 is a constant depending only on t, a, and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22) is not affected by switching regimes, we have that n � k=1 ∥Wn∆Lk(u)∥4 ≤ 32 (n log n)2 �� β(β + 2) �4∥u∥4 + � a(β + 2) �4∥u∥4 C(t, a, β) � → 0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='32) as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' This implies n � k=1 E � ∥Wn∆Lk(u)∥4|Fk−1 � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Therefore, for all ϵ > 0, we obtain n � k=1 E � ∥Wn∆Lk(u)∥21{∥WnLk(u)∥2>ϵ}|Fk−1 � ≤ 1 ϵ2 n � k=1 E � ∥Wn∆Lk(u)∥4|Fk−1 � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' and the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Given the matrix-valued sequence (Wn)n∈N define in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='28), we have ∞ � n=1 1 � log � det W −1 n �2�2 E � ∥Wn∆Ln(u)∥4|Fn−1 � < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='28), it is immediate that det W −1 n = 1 2β + 1 � n log n · anµn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='33) Then, we obtain by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6) that log � det W −1 n �2 log log n → 1 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='34) Hence, there exists a constant C(a, β) > 0 depending only on a and β such that ∞ � n=1 1 � log � det W −1 n �2�2 E � ∥Wn∆Ln(u)∥4|Fn−1 � ≤ ∞ � n=1 C(a, β) (log log n)2 E � ∥Wn∆Ln(u)∥4|Fn−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='35) Hereafter, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='31) together with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='32) imply that ∞ � n=1 1 (log log n)2 ∥Wn∆Ln(u)∥4 ≤ C′(a, β) ∞ � n=1 1 (n log n log log n)2 < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' for some other constant C′(a, β) > 0 depending only on a and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Finally, using the above equation together with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='35) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ 32 JIAMING CHEN AND LUCILE LAULIN Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Fix the test vector u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The growth rate of the compensator of the partial sum of (Nn(u)2)n∈N is less than cubic growth, in the sense that 1 n3 n−1 � k=1 E � Nk+1(u)2|Fn � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The law of iterated expectations and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='8) yields 1 nE � E � Nn+1(u)2|Fn �� = 1 nE � ⟨N(u)⟩n � → � β β − a(β + 1) �2 uT u as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The strong law of large numbers then yields 1 n n−1 � k=1 1 k E � Nk+1(u)2|Fk � → � β β − a(β + 1) �2 uT u as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence 1 n3 n−1 � k=1 E � Nk+1(u)2|Fn � ≤ 1 n2 n−1 � k=1 1 k E � Nk+1(u)2|Fk � → 0 as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The barycenter process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For the following Toeplitz Lemmas, see [18] and [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [33, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1 Part I] Let (an,k)1≤k≤kn, n∈N be a double array of real numbers such that for all k ≥ 1, we have an,k → 0 as n → ∞ and supn∈N �kn k=1 |an,k| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Let (xn)n∈N be a real sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' If xn → 0 as n → ∞, then �kn k=1 an,kxk → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [33, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1 Part II] Let (an,k)1≤k≤kn, n∈N be a double array of real numbers such that for all k ≥ 1, we have an,k → 0 as n → ∞ and supn∈N �kn k=1 |an,k| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Let (xn)n∈N be a real sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' If xn → x as n → ∞ with x ∈ R and �kn k=1 an,k = 1, then �kn k=1 an,kxk → x as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Quadratic rate estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Our first result is about the convergence rate of the process (Yn)n∈N defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all p ∈ (0, 1), then we have, as n → ∞, E[YnY T n ] ∼ n2a(β+1) Γ(1 + 2a(β + 1)) · 1 dId + n1+2β Γ(β + 1)2(1 + 2β − 2a(β + 1))(β + 1) · 1 dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='11) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='13), we see E � Yn+1Y T n+1|Fn � = � 1 + 2a(β + 1) n � YnY T n + µ2 n+1 �a(β + 1) nµn+1 Σn + 1 − a d Id � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, remember that E � Σn � = d � j=1 E � N X n (j) � ejeT j = d � j=1 n � k=1 P � Xj k ̸= 0 � µk · ejeT j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1 yields E[(nµn+1)−1Σn] ∼ (β + 1)−1 · 1 dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, E � Yn+1Y T n+1 � ∼ � 1 + 2a(β + 1) n � E � YnY T n � + µ2 n+1 β + 1 · 1 dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 33 A recursive argument then gives E � YnY T n � ∼ Γ(n + 2a(β + 1)) Γ(n)Γ(1 + 2a(β + 1))E � Y1Y T 1 � + n−1 � j=1 µ2 j β + 1 · �n−1 k=1(1 + k−12a(β + 1)) �j−1 k=1(1 + k−12a(β + 1)) 1 dId ∼ Γ(n + 2a(β + 1)) Γ(n)Γ(1 + 2a(β + 1)) · 1 dId + n−1 � j=1 µ2 j β + 1 · Γ(n + 2a(β + 1))Γ(j) Γ(j + 2a(β + 1))Γ(n) · 1 dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Employing the asymptotics in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ The process Yn = �n k=1 µkXk differs from Sn by a multiplicative factor at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' When there is no amnesia, the asymptotics of these two processes coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' However, when β ≥ 0, we have to treat the general case in another way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all p ∈ (0, 1) and test vector u ∈ Rd, we have, as n → ∞, E � ⟨M(u)⟩n � ∼ wnuT u − (C1n−1 + C2n−2(a(β+1)−β))uT u, and E � ⟨N(u)⟩n � ∼ � β β − a(β + 1) �2 nuT u − (C1n1−2(1−a)(β+1) + C2)uT u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2 E � ⟨M(u)⟩n � = E � Tr⟨M⟩n � uT u = wnuT u − n � k=1 (γk − 1)2a2 k+1uT E � YkY T k � u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14 and a finite summation, E � ⟨M(u)⟩n � ∼ wnuT u − n−1 � k=1 a2(β + 1)2 k2 (k + 1)−2a(β+1)(C1k2a(β+1) + C2k1+2β)uT u ∼ wnuT u − (C1n−1 + C2n−2(a(β+1)−β))uT u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Similarly, E � ⟨N(u)⟩n � = E � Tr⟨N⟩n � uT u = � β β − a(β + 1) �2 nuT u − n−1 � k=1 a2(β + 1)2 k2 µ−2 k+1uT E � YkY T k � u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Hence, using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14 again, we observe E � ⟨N(u)⟩n � ∼ � β β − a(β + 1) �2 nuT u − n−1 � k=1 a2(β + 1)2 k2 (k + 1)−2β(C1k2a(β+1) + C2k1+2β)uT u ∼ � β β − a(β + 1) �2 nuT u − (C1n1−2(1−a)(β+1) + C2)uT u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all p ∈ (0, 1) and test vector u ∈ Rd, we have, as n → ∞, E � ⟨M(u), N(u)⟩n � ∼ β β − a(β + 1) · Γ(β + 1)Γ(a(β + 1) + 1) (1 − a)(β + 1) n(1−a)(β+1)uT u − (C1n−(1−a)(β+1) + C2n(1−a)(β+1)−1)uT u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='7) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='2, for all test vector u ∈ Rd ∆Ln+1(u) = � βµ−1 n+1 β − a(β + 1) �T ϵn+1(u), 34 JIAMING CHEN AND LUCILE LAULIN and therefore, ⟨M(u), N(u)⟩n = n � k=1 β β − a(β + 1)akµ−1 k E � ϵk(u)ϵk(u)T |Fk−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Taking the trace will give us Tr⟨M, N⟩n = β β − a(β + 1) n � k=1 akµk − β β − a(β + 1) n � k=1 akµ−1 k (γk − 1)2∥Yk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Taking the expectation and using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='14 completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moderate deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all p ∈ (0, 1) and for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , d, ��∆M j n �� ≤ � a(β + 1) + 1 � anµn for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='36) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='1), ∆M j n = anY j n − an−1Y j n−1 = anµnXj n − (an − an−1) n−1 � k=1 µkXj k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Since ∥Xk∥ = 1 for eack k ≤ n, then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4), ��∆M j n �� ≤ anµn + (n − 1)(an−1 − an)µn−1 ≤ anµn + a(β + 1)anµn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' And the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all p ∈ (0, 1) and for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , d, ��∆N j n �� ≤ 2a(β + 1) + β β − a(β + 1) for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='6), ∆N j n = βµ−1 n+1 β − a(β + 1)ϵj n+1 = βµ−1 n+1 β − a(β + 1) · � µn+1Xj n+1 + (1 − γn) n � k=1 Xj kµk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Taking absolute value on both sides, and the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all p ∈ (0, 1) and for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , d, ���� 1 √wn ∆M j k ���� ≤ � a(β + 1) + 1 �anµn √wn for each 1 ≤ k ≤ n, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='37) and in the diffusive and critical regime, ���� 1 wn ⟨M j⟩n − 1 ���� ≤ � � � C · n−1 when a < 1 − 1 2(β+1) C · (log n)−1 when a = 1 − 1 2(β+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Dividing by √wn from both sides of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='36), we get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Moreover, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='9), ��⟨M j⟩n − wn �� ≤ n � k=1 (γk − 1)2a2 k+1∥Yk∥2 ≤ C n � k=1 wk k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Dividing both sides by wn and following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='4), the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' For all p ∈ (0, 1) and for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , d, ���� anµn √wn ∆N j k ���� ≤ � 2a(β + 1) + β β − a(β + 1) �anµn √wn for each 1 ≤ k ≤ n, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='38) MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK 35 and in both the diffusive and critical regime, ���� a2 nµ2 n wn ⟨N j⟩n − 1 ���� ≤ � � � C · n−2(1−a)(β+1) when a < 1 − 1 2(β+1) C · (n log n)−1 when a = 1 − 1 2(β+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Dividing by √wn and multiplied by anµu from both sides of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='18), we get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Then, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='10), we make use of the estimates and the inequalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' □ Denote by Φ(·) := (2π)−1/2 � · −∞ e−t2/2 dt the cumulative distribution of the standard normal random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The following lemmas are straightforward derivations from [19, Theorem 1], see also [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' There exists an absolute constant α′(p, β) > 0 depending only on p, β such that for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , d and all 0 ≤ x ≤ α′(p, β) · n−1/2, in the diffusive and critical regime, P(M j n/√wn ≥ x) 1 − Φ(x) = P(M j n/√wn ≤ −x) 1 − Φ(−x) = � � � C · exp � x3 √n + x2 n + 1 √n(1 + 1 2 log n)(1 + x) � when a < 1 − 1 2(β+1) C · exp � x3 √n + x2 log n + ( 1 √log n + 1 2√n log n)(1 + x) � when a = 1 − 1 2(β+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' There exists an absolute constant α′′(p, β) > 0 depending only on p, β such that for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' , d and all 0 ≤ x ≤ α′′(p, β) · n−1/2, in the diffusive and critical regime, P(anµnN j n/√wn ≥ x) 1 − Φ(x) = P(anµnN j n/√wn ≤ −x) 1 − Φ(−x) = � � � C · exp � x3 √n + x2 n2(1−a)(β+1) + 1 √n(n1/2−(1−a)(β+1) + 1 2 log n)(1 + x) � when a < 1 − 1 2(β+1) C · exp � x3 √n + x2 n log n + ( 1 √n log n + 1 2√n log n)(1 + x) � when a = 1 − 1 2(β+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The authors wish to thank Jean Bertoin and Pierre Tarres for numerous discussions and insightful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Baur.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' E, 94 (5): 052134, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Bena¨ım, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Schreiber, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Tarr`es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Generalized urn models of evolutionary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=', 14 (3): 1455–1478, 2004.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=', 129 (11): 4663–4686, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Marcaccioli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Livan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' A P´olya urn approach to information filtering in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=', 10, 745 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' McRedmond, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Wade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' The convex hull of a planar random walk: perimeter, diameter, and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' 23: 1–24, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [37] F.' metadata={'source': 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Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Sto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=', 48: 66–77, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [38] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Pemantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' A survey of random processes with reinforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Surveys 4 (2007), 1–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [39] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Pemantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Vertex-reinforced random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=', 92: 117–136, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [40] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Sch¨utz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Trimper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Elephants can always remember: Exact long-range memory effects in a non- markovian random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Physical review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' E 70, 045101 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Sheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Arzela-Ascoli’s Theorem and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=', 2022100209, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Touati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Sur la convergence en loi fonctionnelle de suites de semimartingales vers un m´elange de mouvements browniens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Veroyatnost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' i Primenen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=', 36 (4): 744–763, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Wade, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Convex hulls of planar random walks with drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=', 143 (1): 433–445, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' Departement Mathematik, ETH Z¨urich Current address: 101, R¨amistrasse, CH-8092 Z¨urich, Switzerland Email address: jiamchen@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='ch Laboratoire de Math´ematiques Jean Leray, Nantes Universit´e Current address: 2 Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content=' de la Houssini`ere, 44322 Nantes, France Email address: lucile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='laulin@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'} +page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}