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+arXiv:2301.03542v1  [math.ST]  9 Jan 2023
+A Sequential Test for Log-Concavity
+Aditya Gangrade1, Alessandro Rinaldo1 and Aaditya Ramdas12
+agangra2@andrew.cmu.edu, arinaldo@cmu.edu, aramdas@cmu.edu
+1Department of Statistics and Data Science, Carnegie Mellon University
+2Machine Learning Department, Carnegie Mellon University
+Abstract
+On observing a sequence of i.i.d. data with distribution P on Rd, we ask the question of how one can
+test the null hypothesis that P has a log-concave density. This paper proves one interesting negative and
+positive result: the non-existence of test (super)martingales, and the consistency of universal inference.
+To elaborate, the set of log-concave distributions L is a nonparametric class, which contains the set G of
+all possible Gaussians with any mean and covariance. Developing further the recent geometric concept of
+fork-convexity, we first prove that there do no exist any nontrivial test martingales or test supermartingales
+for G (a process that is simultaneously a nonnegative supermartingale for every distribution in G), and
+hence also for its superset L. Due to this negative result, we turn our attention to constructing an e-
+process — a process whose expectation at any stopping time is at most one, under any distribution in L
+— which yields a level-α test by simply thresholding at 1/α. We take the approach of universal inference,
+which avoids intractable likelihood asymptotics by taking the ratio of a nonanticipating likelihood over
+alternatives against the maximum likelihood under the null. Despite its conservatism, we show that the
+resulting test is consistent (power one), and derive its power against Hellinger alternatives. To the best of
+our knowledge, there is no other e-process or sequential test for L.
+1
+Introduction
+Log-concavity is an important and prevalent modelling assumption in the modern study of shape-constrained
+nonparametrics [Sam18].
+Log-concave distributions include many common families of densities, including
+normal, exponential, extreme-value, and logistic distributions, and further are frequently justified in diverse
+application domains including economics, reliability theory and filtering in engineering, and survival analysis
+in medicine [BB06]. At the same time, the family is technically amenable, and admits a unique maximum
+likelihood estimate with a well developed minimax theory and computationally efficient estimators [CS10;
+CDSS18; KDR19; Axe+19; DR11; RS19; CSS10].
+As a result, log-concave densities offer practitioners a
+broadly applicable and usable structure.
+Given the attractive properties of estimation within the log-concave family, tests for membership in the
+same are an important and necessary line of investigation. We note that along with the applications mentioned
+above, such tests also have theoretical interest; for instance, in much of computational learning theory, efficient
+learning algorithms are only known when covariates are sampled according to a log-concave distribution [e.g.
+KKMS08]. While the estimation of log-concave densities has seen significant advances over the past decade
+or two (see, e.g., the citations above, and the survey by Samworth [Sam18]), testing for log-concavity has
+been relatively poorly developed. Indeed, prior to 2021, there were no valid and powerful tests for the same—
+both theoretically and practically—outside of certain restricted one-dimensional settings.
+In a significant
+development, recent work of Dunn et al.
+[DGWR21] has developed such a test, based on the Universal
+Inference strategy of Wasserman et al. [WRB20].
+Our work is concerned with testing log-concavity in a sequential setting. Concretely, we assume that we
+are given streaming access to a sequence {Xt} that are drawn independently and identically from some d-
+dimensional density p, and we wish to test the membership of p within the family of log-concave densities.
+Such a sequential test can be identified with a stopping time τ, where stoppage indicates rejection of the null
+1
+
+hypothesis, and the test is α-valid if under the null, the probability that τ < ∞ is bounded by α. The principal
+attractiveness of such sequential tests arises from their adaptivity: rather than fixing a number of samples a
+priori, the test may adapt to the difficulty of the underlying instance, rejecting earlier in easier settings, and
+allowing for a greater number of samples to detect subtle deviations from the null hypothesis.
+Below, we first set up some notation, and then proceed to contextualise our study, and give a brief overview
+of the contributions of our paper.
+1.1
+Problem setup and background
+We begin by describing the notation needed for our discussion, the testing problem under consideration, and
+the fundamental notions of test martingales and e-processes. We shall give further definitions and details in
+§2, as well as later in the text as the context arises.
+Spaces and measures. Let {Xt} = (X1, X2, . . . ) denote a sequence of d-dimensional random vectors with entries
+indexed by t, which are measurable maps from Ω := (Rd)N to Rd, endowed with the cylindrical Borel sigma-
+algebra B(Rd)N. We use typewriter style fonts, e.g. P, to denote laws of random processes (i.e. probability
+measures on (Ω, B(Rd)N)), and standard fonts, e.g. P to denote laws on (Rd, B(Rd)). We use F = {Ft}
+to denote the natural filtration of the process {Xt}, where Ft := σ(X1, . . . , Xt), for each t. For a Borel
+probability measure P on Rd, we use P ∞ to denote the law of an i.i.d. process drawn according to P. We
+use D to denote the set of probability measures on Rd with Lebesgue densities, and D∞ = {P ∞ : P ∈ D}.
+For P ∈ D, we use p to denote its Lebesgue density. For technical convenience we define D1 := {P ∈ D :
+E[max(0, log p(X))] < ∞, E[∥X∥] < ∞}. A set of laws P is said to be mutually absolutely continuous (m.a.c.)
+if for all P, Q ∈ P, P ≪ Q ≪ P. Finally, we frequently use Xt
+1 := (X1, . . . , Xt) to denote finite prefixes of
+{Xt}.
+Log-concave measures. A function f : Rd → R is said to be log-concave if there exists a concave function
+g such that f = eg. If f is further a density with respect to the Lebesgue measure, then it is said to be a
+log-concave density. We denote the set of measures with log-concave densties as L, and use L∞ to denote the
+set of i.i.d. log-concave measures on euclidean sequences, i.e. L∞ := {P ∞ : P ∈ L}.
+Sequential test for log-concavity.
+The testing problem of interest is formulated as follows: Let Xt
+i.i.d.
+∼ P
+for some unknown P ∈ D. We wish to test the null hypothesis H0 : P ∈ L.
+A sequential test corresponds to {Ft}-adapted stopping time, representing the (possibly infinite) time at
+which the test stops and rejects the null hypothesis. We shall refer to this stopping time as the rejection time
+of the sequential test. A test is said to be α-valid if its rejection time, τ satisfies that
+sup
+P ∈L
+P ∞(τ < ∞) ≤ α,
+meaning that, under the null, the probability of ever rejecting, i.e. of incurring a Type I error, is at most α.
+Similarly, a test is said to be asymptotically (1 − β)-powerful against Q ⊂ D \ L if the probability of failing to
+reject the null under any distribution in the alternative Q (also know as a type II error) is uniformly bounded
+by β:
+inf
+Q∈Q Q∞(τ = ∞) ≤ β.
+A test is said to be consistent against Q if it is asymptotically 1-powerful against the same. Note that, when
+consistent, these tests are typically called ‘power-one tests’ (following Robbins) to differentiate them from the
+traditional Waldian sequential testing paradigm for which stopping does not imply rejection of the null.
+Test martingales, test supermartingales and e-processes.
+We briefly survey key notions underlying
+our discussion, namely test martingales, and e-processes, leaving details to §3 and §4 respectively.
+Definition A process {Mt} is a nonnegative supermartingale (NSM) with respect to a filtration {Ft} and
+a law P if it is adapted, nonnegative, and EP[Mt|Ft−1] ≤ Mt−1 for each t. If the inequality is further an
+equality at each t, then {Mt} is a nonnegative martingale (NM). We shall succinctly say that such a process
+is a P-NSM or P-NM respectively.
+2
+
+Obviously, every P-NM is also a P-NSM. An important basic inequality of Ville [Vil39] controls the tail
+behaviour of NSMs: if {Mt} is a P-NSM such that M0 = 1, then for every α ∈ (0, 1],
+P(∃t ≥ 1 : Mt ≥ 1/α) ≤ α.
+The result above is a sequential (time-uniform) analogue of Markov’s inequality. Equivalently, one can make
+claims at arbitrary stopping times: for all stopping times τ, P(Mτ ≥ 1/α) ≤ α. This can be seen by applying
+the optional stopping theorem for NSMs [Mey66, Ch. V, Thm. 28] and Markov’s inequality.
+We now extend the above notions to composite families of sequential laws. Throughout this paper we shall
+take the filtration to be the natural filtration of the data, and will leave it implicit in our definitions below.
+Definition For a set of sequential laws P, we say that a process {Mt} is a P-NSM if {Mt} is a P-NSM for
+every P ∈ P. Similarly, {Mt} is a P-NM if it is a P-NM for every P ∈ P. A P-NSM such that M0 = 1 is
+called a test supermartingale for P, and a P-NM such that M0 = 1 is called a test martingale for P.
+Observe that test supermartingales satisfy Ville’s inequality for each P ∈ P, i.e., if {Mt} is a test super-
+martingale for P, then for every α ∈ (0, 1],
+∀P ∈ P, P(∃t ≥ 1 : Mt ≥ 1/α) ≤ α.
+(1)
+Test supermartingales are so named because they form the canonical path to sequentially testing composite
+hypotheses, which is encapsulated entirely by the above relation, in that valid tests can be derived by rejecting
+only when a test supermartingale crosses a threshold. They are particularly interesting in nonparametric
+settings; for example one can use them to sequentially test the mean of a bounded random variable [WR23],
+for testing symmetry [RRLK20], for two-sample testing [SR21], independence testing [PBKR22], and testing
+calibration [AHZ21], to mention only a few interesting sequential nonparametric problems. We shall discuss
+test supermartingales extensively in this paper.
+E-Processes [RGVS22; RRLK20; GHK19; HRMS20] are a recently defined class of processes that will also
+play a central role in this paper.
+Definition A process {Et} is called an e-process with respect to a sequential law P if it is non-negative, and
+for every stopping time τ, we have EP[Eτ] ≤ 1. Similarly, {Et} is an e-process for a class of sequential laws P
+if it is an e-process with respect to every P ∈ P.
+E-Processes have a variety of equivalent definitions [RRLK20, Lem. 6]. In particular it is sufficient for the
+process to satisfy EP[Eτ] ≤ 1 for only bounded stopping times.
+By the optional stopping theorem (which holds without restriction on stopping times for nonnegative
+supermartingales), notice that every test supermartingale for a class P is also an e-process for this class.
+Thus, e-processes generalise the notion of test supermartingales. We observe that a Ville-type relation also
+holds for e-processes, simply due to Markov’s inequality: if Et is an e-process for P, then for every α ∈ (0, 1],
+∀P ∈ P, stopping times τ, P(Eτ ≥ 1/α) ≤ αE[Eτ] ≤ α.
+(2)
+Much as Ville’s inequality over the class (1) captures the relevance of test supermartingales to sequential
+testing, the above inequality captures the relevance of e-processes to the same. The notion of e-processes,
+along with the non-sequential analogue of e-values, is gaining vogue in recent work in statistics due to this key
+property, along with the fact that e-processes exist for many composite and nonparametric testing problems
+for which test supermartingales do not exist (see, e.g., the recent survey by Ramdas et al. [RGVS22]). We
+will also encounter this situation in the current paper.
+It is important to note that test supermartingales or e-processes can directly be interpreted as evidence
+against the null hypothesis: since we expect them to be less than one under the null, the larger their realized
+value, the more evidence we have that the null hypothesis is wrong.
+Thus, there is no explicit need to
+threshold them at 1/α for some prespecified α; one can alternatively simply report the final value at the final
+stopping time of the experiment (which can itself be arbitrarily chosen). Nevertheless, we present this paper
+in the language of level-α tests because that is far more popular, and we refer the interested reader to the
+aforementioned references for further discussion on e-processes.
+3
+
+1.2
+Inadequacy of Test (Super)Martingales, and the Power of E-Processes
+One dominant (but sometimes hidden) principle behind sequential testing of composite hypotheses is the use
+of nonnegative martingales (NMs), or nonnegative supermartingales (NSMs). Concretely, to test a composite
+hypothesis P ∈ P, one attempts to construct a P-test supermartingale {Mt}, which was defined earlier. By
+Ville’s inequality (1), the chance that Mt ever exceeds 1/α under any null law is bounded by α. Thus, these
+test supermartingales immediately yield a valid test: reject the null when Mt ≥ 1/α. The associated rejection
+time, of course, is the Mt-hitting time of 1/α. Such tests have game-theoretic interpretations, through the fact
+that nonnegative (super)martingales represent wealth processes in betting games [RGVS22]. For example, a
+P-test martingale is the wealth process of a gambler who bets against the hypothesis that the sequence {Xt}
+is drawn according some law in P. The game is designed so that the gambler cannot hope to reliably (in
+expectation) make money if the null hypothesis is true; this is imposed by a restriction that under any law in
+P, the expected wealth multiplier in each round should be at most unity.
+However, for sufficiently rich classes P, such a game leaves the gambler powerless; the gambler is so
+constrained by the aforementioned restriction that the only option is to not bet at all (or throw away money).
+This phenomenon was first observed in work on testing exchangability in discrete time binary processes by
+Ramdas et al. [RRLK22], who demonstrated that any process {Mt} that is an NSM for all exchangable binary
+laws is, almost surely, a strictly decreasing process (the wealth starts at one and can only possibly go down).
+As a result, any test based on thresholding such processes must be powerless against any alternative. Our
+first technical contribution demonstrates an anlogous phenomenon in the setting of log-concave distributions.
+Specifically, we show that the smaller class of i.i.d. Gaussian processes is not testable using NMs (or NSMs),
+since all such processes are trivial in the sense of being almost surely constant (or decreasing). The claim is
+summarised below, where G∞ denotes the set of all i.i.d. Gaussian laws (of any mean and variance).
+Theorem. (Informal) There are no nontrivial G∞-NSMs or G∞-NMs. A fortiori, there are also no nontrivial
+L∞-NSMs or L∞-NMs.
+Thus, log-concave densities represent a natural class of distributions that cannot be tested via martingales.
+Testing via E-Processes.
+Given that one cannot test for log-concavity (or indeed, Gaussianity) using
+nonegative (super)martingales, we are left in a situation where the prevalent design paradigm for sequential
+testing is neutralised. There are two contrasting lines of attack that can be employed instead.
+The first of these involves designing a restricted filtration Gt, distinct from the natural filtration, under
+which there might exist nontrivial test supermartingales. Ramdas et al. [RRLK22; RGVS22] highlight the
+remarkable fact that shrinking a filtration could introduce new nontrivial (composite) test martingales when
+none existed in the original filtration. Such a strategy was notably used by Vovk et al. [VNG03; FGNV12] to
+develop a sequential test for exchangeability, where as mentioned above, no nontrivial test supermartingales
+exist in the data filtration. There are two main disadvantages to such an approach. First, such test martingales
+only yield an e-process for a restricted set of stopping times (those under the restricted filtration). From an
+applied point of view, the use of such an e-process demands discipline from a practitioner—they cannot look
+at the raw data to decide when to adaptively stop (a predefined stopping rule, like the hitting time of 1/α
+is okay, but it may never be reached, in which case we may still wish to present the obtained evidence at
+the stopping time). Second, from a design point of view, the construction of appropriate filtrations is itself a
+subtle task that is heavily problem-dependent, and thus designing such tests is more of an art than a science.
+In particular, no such construction is known or obvious for sequential log-concavity testing.
+In contrast, we follow the alternative strategy of testing via an e-process. Recall that a process {Et} is
+an e-process for a set of sequential laws P if, for every stopping time τ and every P ∈ P, EP[Eτ] ≤ 1. Such
+processes bear a deep relationship to the aforementioned test martingales. Indeed, it has been argued that
+(admissible) e-processes must take the form infP∈P M P
+t , where each {M P
+t } is a P-NM [RRLK20]. The same
+observation lends e-processes a gambling interpretation as the wealth process of a gambler against a ‘family of
+games’, wherein the gambler simultaneously plays a game against each P ∈ P, and their wealth is taken as the
+smallest wealth amongst these games. The gambler can then make money only if each of these games makes
+money, i.e., if ∀P ∈ P, M P
+t grows without bound, which would then indicate that every P ∈ P can be rejected.
+E-Processes offer a similar testing approach as the previously discussed test supermartingales, as elucidated
+4
+
+by the inequality (2). Indeed, given an e-process {Et} for P, we can construct an α-valid test of membership
+in P by rejecting only if Et ≥ 1/α. Indeed, in this case, the rejection time is
+τα := inf{t ≥ 1 : Et ≥ 1/α},
+and using the inequality (2), we may conclude that
+∀P ∈ P, P(τα < ∞) ≤ α,
+i.e. this test is valid for the composite null P. Note further that the validity extends beyond this: let σ be any
+other stopping time with respect to the natural filtration of the data. We further have that P(Eσ ≥ 1/α) ≤ α,
+and thus no extraneous stopping criterion can affect the validity of the test, as long as rejection occurs only
+if Eσ ≥ 1/α.
+The theory and applications of e-processes have seen considerable development in the recent literature
+on sequential analysis (along with the more basic notion of e-variables in batched settings). The concept is
+attractive thanks to its flexibility and simplicity (despite generalizing nonnegative martingales), but construct-
+ing powerful e-processes is partly science and partly art [RGVS22]. In composite testing, e-processes are of
+central importance since they do not encounter the same pitfalls as NSMs and NMs, and there do indeed exist
+nontrivial e-processes even on classes where no such NSMs exist. Indeed, in some sense, e-processes can be
+shown to lie at the very core of sequential composite testing [RLKR22].
+1.3
+Test Using Universal Likelihood Ratios: A simple E-Process
+The universal inference strategy [WRB20] gives a simple and generic construction of e-processes when a
+maximum likelihood estimate can be easily computed.
+To contextualise this approach, we first consider the case of a point null and alternative P ∞ and Q∞. In
+this case, classical sequential testing theory posits that the sequential likelihood ratio
+Lt =
+t�
+s=1
+q(Xs)
+p(Xs)
+yields a valid and powerful test upon thresholding at 1/α. Indeed, under the null, {Lt} is an e-process, since
+it is an NM.
+Against simple nulls but composite alternatives, likelihood ratios such as the above are typically adjusted
+to account for the variety of possible alternatives. One way to do this is to replace the above numerator with an
+estimate ˆqs(Xs). Importantly, as long as this ˆqs is nonanticipating, i.e., is Ft−1-measurable (depending only
+on the first t − 1 datapoints), the martingale property continues to hold. To highlight this nonanticipation,
+we shall denote these estimators as ˆqs−1. A second option is to mix over alternatives, perhaps using some
+non-informative “prior”, but we will go with the first option in this paper because we are dealing with a highly
+nonparametric alternative (essentially the complement of all log-concave laws, or the unspecified subset of
+those against which one may hope to have power) — it is easy to use kernel density estimates for ˆqs, but not
+so easy to mix over such a loosely specified nonparametric alternative.
+The sequential universal likelihood ratio statistic (ULR) extends the above to composite nulls when a
+maximum likelihood estimator (MLE) is computable.
+Concretely, the statistic is as follows: let ˆqt−1 be
+any predictable probability density, that is ˆqt−1 may be expressed as a function of only {X1, . . . , Xt−1} and
+additional independent randomness. As before, we should think of ˆq as trying to estimate the underlying law
+p. Let ˆpt be the MLE over the null class L with the data Xt
+1, i.e.,
+ˆpt = arg max
+ˆp∈L
+�
+s≤t
+log ˆp(Xs).
+Notice that, unlike ˆqt−1, the MLE ˆpt makes use of Xt. The sequential ULR statistic is the process
+Rt :=
+�
+s≤t
+ˆqs−1(Xs)
+ˆpt(Xs) .
+5
+
+(Of course, if the numerator was simply �
+s≤t ˆqt(Xs), where ˆqt is an MLE over a larger class calculated using
+{X1, . . . , Xt}, then we would get the usual generalized likelihood ratio process.
+However, we will handle
+very rich nonparametric alternatives over which computing the MLE is for all practical purposes impossible,
+and further, for irregular models like log-concave distributions, such generalized likelihood ratios are very
+ill-behaved and not well understood.)
+The principal factor underlying the utility of Rt is that it is an e-process. Indeed, for any P ∈ L, and t,
+Rt is dominated by Ft(P) = �
+s≤t ˆqs−1(Xs)/p(Xs). Further, {Ft(P)} is a P ∞-martingale started at 1, due to
+the predictability of ˆq, and thus for any stopping time τ,
+EP ∞[Rτ] ≤ EP ∞[Fτ(P)] ≤ 1.
+Notice in the argument above that while the e-process is dominated by a P ∞-martingale, it is not itself a
+martingale. Indeed, this property is crucial to the existence of nontrivial e-processes even when there are no
+such test martingales. We note that this property of domination by a P ∞-NM for every P ∈ L (or in general
+P-NM for P ∈ P) is equivalent to the e-process property itself, and can be taken as an alternate definition of
+the same [RRLK20, Lem. 6].
+Due to the above observation, the ULR e-process yields a valid test upon thresholding at 1/α. The power
+of any such test relies on the two aspects of how well ˆpt and {ˆqs}s≤t estimate the underlying law p. Indeed,
+we argue in §4 that if p ̸∈ L, then �
+s≤t p(Xs)/ˆpt(Xs) must grow exponentially with t. Thus, as long as
+the sequential estimates ˆqt approximate p well in a cumulative regret sense, the procedure above must be
+consistent. Concretely, define the regret of prediction using {ˆqt} as
+ρt(ˆq; P) :=
+�
+s≤t
+(− log ˆqs−1(Xs)) −
+�
+s≤t
+(− log p(Xs)),
+so that better estimation results in lower regret, and define the ‘well-estimable’ class
+Q(ˆq) := {P : ρt(ˆq; P)/t → 0 P-a.s. as t ր ∞}.
+In Section 4 we show the following:
+Theorem. (Informal) Let Rt denote the ULR e-process with the sequential estimator E . Then the test that
+rejects when Rt ≥ 1/α is α-valid, and consistent against Q(ˆq).
+In fact, in §4, we demonstrate a more refined version of the above statement, which allows ρt to grow
+linearly, but at a rate bounded by the distance of p from log-concavity. In any case, we comment that the
+class Q(·) above is quite rich. For instance, using sieve estimators yields low-regret estimation in the above
+log-loss sense for nonparametric classes such as laws on compact intervals with smooth and bounded densities.
+The ULR e-process thus gives a powerful test for log-concavity against a rich set of alternates, even though
+no test martingale can deliver such properties. Our work thus offers further insight into the sequential testing
+of rich composite nulls, and the primacy that e-processes must take in the modern study of the same.
+Along with the above asymptotic consistency result, we further derive finite rejection rate bounds by
+controlling the typical rejection time of the ULR e-process in terms of the Hellinger distance of the alternaive
+law from log-concavity. In particular, we show explicit bounds on typical rejection times against Lipschitz and
+bounded laws on the unit box. The above theoretical exploration is augmented with simulation studies on a
+simple parametric family comprising a mixture of two Gaussians to empirically evaluate the validity and power
+of the test. We find that in small dimensions d ≤ 3, the tests show excellent validity, as well as reasonable
+power. We further use this simulation study to highlight the role of the quality of the estimators ˆqt in the
+power of the test.
+Summary of Contributions
+To summarise, this paper is concerned with the theoretical and methodolog-
+ical aspects of sequential testing for log-concavity. We first show a negative result that demonstrates that the
+approach of constructing test (super)martingales is powerless for testing this class of laws, and along the way
+also offering simple characterisations of the fork-convex hull of i.i.d. sequential laws. In the positive direction,
+we propose using the Universal Inference based e-process as a way to test log-concavity in the absence of test
+martingales. We theoretically demonstrate both the consistency of the resulting sequential test, along with
+concrete adaptive bounds on typical rejection time under a wide class of alternatives, and illustrate the same
+via simulation studies.
+6
+
+2
+Definitions, and Background on Log-Concave Distributions
+We begin with basic background on log-concave distributions, and necessary notation. We refer the reader to
+the survey of Samuard and Wellner for further details [SW14].
+Log-Concave Laws.
+A distribution P on (Rd, B(Rd)) is called logarithmically concave (henceforth log-
+concave) if for every pair of compact sets A, B and λ ∈ (0, 1),
+P(λA + (1 − λ)B) ≥ P(A)λP(B)1−λ,
+λA + (1 − λ)B is the Minkowski sum {λx + (1 − λ)y : x ∈ A, y ∈ B}. It is well known that a distribution
+that admits a density with respect to the Lebesgue measure is log-concave if and only if P(dx) = eg(x)dx for
+a concave function g. Recall that L denotes the class of log-concave distributions with density on Rd, while
+L∞ denote the set of i.i.d. sequential laws P ∞ for P ∈ L.
+Log-Concave M-projection.
+Recall that D denotes the set of laws on (Rd, B(Rd)) that admit densities
+with respect to the Lebesgue measure, and that
+D1 := {P ∈ D : E[max(0, log p(X))] < ∞, E[∥X∥] < ∞},
+where p(·) is the density of P. For every P ∈ D1, there exists a unique law
+LP := arg min
+L∈L
+KL(P∥L),
+where KL(·∥·) is the KL-divergence, called its log-concave M-projection. We shall abuse notation and use Lp
+to denote the Lebesgue density of LP (one is admitted as long as P ∈ D1). For a set of points {xt
+1}, t ≥ d + 1,
+the log-concave maximum likelihood estimator (MLE) is the log-concave M-projection of the empirical law
+Pt = �
+s≤t δxs/t, denoted ˆPt. Most commonly, we shall refer to its Lebesgue density, ˆpt, which may equivalently
+be defined as
+ˆpt :=
+arg max
+log f is a concave function
+f≥0,
+�
+f=1
+�
+s≤t
+log f(xs).
+The log-concave MLE has extremely favourable theoretical properties when Xt
+1
+i.i.d.
+∼ P for some P ∈ D1. For
+instance ˆpt → Lp in the strong sense that ∃a > 0 :
+�
+ea∥x∥|ˆpt(x) − Lp(x)| → 0 almost surely [CS10].
+Locally Absolutely Continuous Sequential Measures.
+Let Γ denote the standard Gaussian law on Rd,
+γ its density, and let Γ = Γ∞. Notice that Γ is the law of a white noise. A sequential law P is said to be
+locally absolutely continuous (l.a.c.) with respect to Γ, denoted P ≪loc. Γ, if for all t, the law of the finite
+prefix P|t(·) := P(Xt
+1 ∈ ·) is absolutely continuous with respect to Γ|t. Such l.a.c. laws admit a density process,
+denoted
+ZP
+t := dP|t
+dΓ|t
+.
+As an example, if P = P ∞ for some law P with Lebesgue density p, then P ≪loc. Γ, and ZP
+t = �
+s≤t p(Xs)/γ(Xs).
+Of course, we may specify sequential laws (that are ≪loc. Γ) by specifying their density processes. Note that
+ZP
+t is a likelihood ratio process with respect to Γ, and so is a Γ-martingale. We shall henceforth use Γ as a
+reference measure for sequential laws, and almost entirely work under laws that are ≪loc. Γ.
+Notice that if ZP
+t−1 > 0 then for {Xt} ∼ P, ZP
+t /ZP
+t−1 is the Γ-conditional density of Xt given Xt−1
+1
+. Further,
+ZP
+t−1 = 0 =⇒ ZP
+t = 0. As a result, we may write for any adapted process {Mt} that
+ZP
+t−1EP[Mt(Xt
+1)|Ft−1] = ZP
+t−1EP[Mt(Xt
+1)1{ZP
+t > 0}|Ft−1] = EΓ[MtZP
+t 1{ZP
+t > 0}|Ft−1] = EΓ[MtZP
+t |Ft−1].
+From this, we observe that a process {Mt} is a P-NSM if and only if {ZP
+t Mt} is a Γ-NSM. Indeed, if the
+former is true, then we conclude from the above that EΓ[MtZP
+t ] ≤ ZP
+t−1Mt−1, while if the latter is true, then we
+can conclude that ZP
+t−1EP[Mt|Ft−1] ≤ ZP
+t−1Mt−1 ⇐⇒ 1{ZP
+t−1 > 0}EP[Mt|Ft−1] ≤ 1{ZP
+t−1 > 0}Mt−1. Since
+1{ZP
+t−1 > 0} holds P-a.s., it follows that EP[Mt|Ft−1] ≤ Mt−1 P-a.s., and so Mt is a P-NSM. By maintaining
+equalities in the above analysis, the analogous statement also holds for NMs. These facts are quite useful in
+our later study of fork-convex hulls.
+7
+
+3
+There Are No Nontrivial Test Supermartingales for Log-concavity
+We begin with defining a natural notion of triviality.
+Definition An NSM {Mt} is said to be trivial if Γ(∃t : Mt > Mt−1) = 0. An NM {Mt} is said to be trivial if
+Γ(∃t : Mt ̸= Mt−1) = 0.
+In words, a NSM (NM) is trivial if, almost surely, it is a non-increasing (constant) process. For the remain-
+der of this section, we will set {Ft} to be the natural filtration. We recall the notion of test supermartingales
+for a class of laws P, which we shall refer to as just nonnegative supermartingales.
+Definition For a set sequential laws P, we say that a process {Mt} is a P-NSM if {Mt} is a P-NSM for every
+P ∈ P. Similarly, {Mt} is a P-NM is it is a P-NM for every P ∈ P.
+With these definitions in hand, we state the main result of this section, the proof of which is left to §3.3.
+Theorem 1. There are no nontrivial G∞-NSMs or G∞-NMs under the natural filtration, and a fortiori, there
+are no nontrivial L∞-NSMs or L∞-NMs under the natural filtration.
+As discussed by Ramdas et al. [RRLK22], the above result implies that any valid level-α sequential test
+for log-concavity based on thresholding L∞-NSMs or L∞-NMs must be powerless. Indeed, in the former case,
+such a test against any law that is locally absolutely continuous with respect to Γ will almost surely never
+exceed its starting value, and thus will almost surely never reject.
+Intuition behind the proof. The result arises from a contradiction. To illustrate this, suppose {Mt} is a G∞-
+NSM, and that for some time t, given Ft−1, it increases on the event {Xt ∈ O} for some open ball O, i.e.,
+conditionally on Xt−1
+1
+= xt−1
+1
+, {Xt ∈ O} ⊂ {Mt > Mt−1}. Notice that due to nonnegativity, at worst it could
+be zero outside the ball. Now, consider a Gaussian GO of such a small variance that GO(O) ≈ 1. By tuning
+this variance, we can ensure that Mt > Mt−1 with probability arbitrarily close to 1 given the history, and
+since the drop in the Mt remains bounded outside of the ball, this ensures that the conditional expectation of
+Mt strictly increases. Since this violates the supermartingale property against G∞
+B ∈ G∞, we must conclude
+that no such ball O exists.
+Of course, the set on which {Mt} increases need not contain any ball, but still be of nontrivial mass, not
+to mention that this set may vary with the history in a complex way. We address such gaps by exploiting the
+notion of fork-convexity [RRLK22] which serves as a sequential analogue of convexity especially germane to
+(super)martingale properties, and is treated in the following section. In particular, it holds that any process
+{Mt} that is a G∞-NSM (or NM) is also a NSM (or NM) with respect to any sequential law in the ‘fork-convex
+hull’ of G∞. The main argument then demonstrates that the fork-convex hull of G∞ is incredibly rich, and
+contains the laws of arbitrary independent processes with density (i.e., processes of jointly independent {Xt}
+such that Xt ∼ pt ∈ D). This large set of laws entirely obstructs the NSM (or NM) property from holding
+in any nontrivial manner, essentially using a robust version of the previous intuitive example. Schematically,
+we take the following route to establish this result, where the forward direction of each implication exploits
+fork-convex combinations (and the reverse is trivial).
+G∞-NSM
+G∞
+∗ -NSM
+(
+� G∗)-NSM
+(
+� G∗)-NSM
+(
+� D)-NSM
+Trivial
+⇐⇒
+⇐⇒
+⇐⇒
+⇐⇒
+⇐⇒
+Figure 1: Schematic view of the argument. G∗ is the set of all finite mixtures of Gaussians and G∗ denotes its
+L1 closure. For any set P, the class � P consists of independent sequential laws with marginals in P. See
+§3.2 for definitions.
+3.1
+Fork-convex Combinations
+In an algorithmic sense, for two laws P, Q, an α-convex combination R = αP +(1−α)Q is the law of the output
+of the following procedure: independently sample U ∼ P and V ∼ Q, and output X = U or V according to
+the outcome of an independent α-coin. Fork-convex combinations are the natural sequential extension of such
+8
+
+a procedure. Concretely, we sample two trajectories {Ut} ∼ P and {Vt} ∼ Q, release Xt = Ut for t ≤ s for
+some time s, and then flip a h-coin (where h can depend on the history) to decide whether the subsequent tail
+is Xt = Ut or Xt = Vt for t > s. Notice that this is a much richer notion than convex combinations: firstly,
+the decision to release Ut or Vt only needs to be made for a tail of the output sequence, and secondly, the
+mixture proportion can depend on the history. Formally, this is defined as follows.
+Definition ([RRLK22]) Let P, Q ≪loc. Γ be sequential laws. Let s ∈ N, and let h ∈ [0, 1] be an Fs-measurable
+random variable such that Γ(h < 1, ZQ
+s = 0) = 0. Then the (s, h)-fork-convex combination of P with Q is the
+sequential law R with density process
+ZR
+t := ZP
+t 1{t ≤ s} +
+�
+hZP
+t + (1 − h)ZQ
+t
+ZP
+s
+ZQs
+�
+1{t > s}.
+We shall denote this succinctly as R =
+�
+P
+s,h
+−→ Q
+�
+.
+Notice that fork-convex combinations probabilistically allow single data-dependent change-points, or ‘switches’,
+from a law P to Q. The ratio ZP
+s/ZQ
+s accounts for the fact that the prefix up to time s was drawn according to
+P in the case of a switch, and the condition on h ensures that ZQ
+s ̸= 0 when we switch to Q (informally meaning
+that the initial segment of data was not impossible under Q).
+The importance of the above definition lies in the fact that fork-convex combinations preserve (super)martingale
+properties. Recall from §2 that {Mt} is a P-NSM if and only if {ZP
+t Mt} is a Γ-NSM. Now suppose {Mt} is
+both a P-NSM and Q-NSM, let R be a (s, h)-fork-convex combination of P and Q. For t ≥ s + 1, and we have
+EΓ[ZR
+t Mt|Ft−1] = hEΓ[ZP
+t Mt|Ft−1] + (1 − h)ZP
+s
+ZQs
+EΓ[ZQ
+t Mt] ≤ hZP
+t−1Mt−1 + (1 − h)ZP
+s
+ZQs
+ZQ
+t−1Mt−1 = ZR
+t−1Mt−1,
+where we have utilized the fact that h and Z·
+s are Ft−1-measurable. The same calculation is trivial for t ≤ s,
+and follows similarly for the martingale property.
+The same property extends considerably beyond finite
+combinations to closed fork-convex hulls, which generalise the standard notion of closed convex hull of sets.
+Definition ([RRLK22]) A set is said to be fork-convex if it contains all fork-convex combinations of its
+elements.
+Let P be a set of sequential laws that are locally absolutely continuous with respect to Γ.
+The
+fork-convex hull of P, denoted f-conv(P) is the intersection of all fork-convex sets containing P. The closed
+fork-convex hull of P, denoted f-conv(P) is the closure of its fork-convex hull with respect to L1(Γ) convergence
+of the likelihood ratio processes at every fixed time t.
+Explicitly, the closure in the definition includes all processes Q such that there exists a sequence Qn with
+density process {ZQn
+t } such that ∀t, ZQn
+t
+→ ZQ
+t in L1(Γ). We shall refer to this as the local L1(Γ) closure. This
+closure induces considerable flexibility into closed fork-convex hulls, making the notion a powerful concept
+in light of the following phenomenon, observed by Ramdas et al. [RRLK22, Thm. 13] whose argument we
+reproduce below.
+Proposition 2. For a set of sequential laws P, a process is a P-NSM if and only if it is a f-conv(P)-NSM.
+Proof. The result is evident for the fork-convex hull as an extension of the previous two-point calculation.
+This extends to closures as follows. Let {Mt} be the process in question, and suppose Pn → P in the sense
+above for Pn ∈ f-conv(P). Let Zn
+t := ZPn
+t
+and Zt := ZP
+t . We know that for each t, Zn
+t → Zt in L1(Γ). We need
+to show that ZtMt is a Γ-NSM. To this end, fix a t, and, by passing to a subspace, assume that Zn
+t → Zt and
+Zn
+t−1 → Zt−1 pointwise a.s. Now, since Zn
+t Mt is a Γ-martingale, using Fatou’s lemma yields
+EΓ[ZtMt|Ft−1] = EΓ[lim inf Zn
+t Mt|Ft−1] ≤ lim inf EΓ[Zn
+t Mt|Ft−1] ≤ lim inf Zn
+t−1Mt−1 = Zt−1Mt−1.
+It is worth noting that while the NSM property is preserved under closures above, the same is not necessarily
+true of the martingale property due to the use of Fatou’s Lemma when handling closures in the above proof.
+Nevertheless, the NM (and indeed the martingale property without appeal to non-negativity) persists under
+fork-convex hulls, without the closure, giving us the following characterisation.
+Proposition 3. For a set of sequential laws P, a process is a P-NM if and only if it is a f-conv(P)-NM.
+9
+
+3.2
+The Fork-Convex Hull of Independent Sequential Laws
+Proposition 2 gives us a concrete attack to showing the triviality of G∞-NSMs: we shall show that the fork-
+convex hull of this set is far too rich to allow the existence of nontrivial NSMs. The bulk of our argument
+develops simple structural characterisations of fork-convex hulls of independent sequential laws. This section
+describes this characterisation using three properties, whose proof we leave to §A.2. We begin with a key
+definition that sets notation for ‘independent sequential laws’ from a set.
+Definition Let P be a set of distribution on Rd. For a sequence of distributions {Pt}t∈N, we define �{Pt}
+as the sequential distribution of a stochastic process {Xt}t∈N such that all Xt are jointly independent, and for
+each t ∈ N, Xt ∼ Pt. We further define � P := {�{Pt} : Pt ∈ P ∀t}, i.e. the set of laws of independent
+stochastic processes with laws at each time lying in P.
+Note that � P is a much richer set than the i.i.d. sequential laws, which we denote P∞ := {P ∞ : P ∈ P}.
+In light of this, the following result demonstrates the richness of fork-convex hulls. Recall that a set of laws is
+mutually absolutely continuous (m.a.c.) if every pair of laws contained in it is mutually absolutely continuous.
+Lemma 4. Let P ⊂ D be a m.a.c. set of laws with density on Rd. Then, f-conv(P∞) ⊃ � P ⊃ P∞.
+To sketch the argument underlying the above, fix any P = �{Pt}. It suffices to demonstrate a sequence of
+laws {RT}T ∈N, each generated by finite fork-convex combinations of P∞-laws (and their fork-convex combina-
+tions) such that for t ≤ T, the density process of RT and P agree. The conclusion then follows under closure,
+since RT → P in the appropriate sense. The concrete witness for the above Lemma is the following sequence
+R1 := P ∞
+1 , RT =
+�
+RT −1 T −1,0
+−→ P ∞
+T
+�
+,
+where each fork-convex combination is valid since P is m.a.c. In essence, this exploits the fact that fork-convex
+combinations let one switch between laws after a time of our choosing. See A.2 for details.
+Next, we exploit the convex combination properties of fork-convex combinations to demonstrate that fork-
+convex combinations of i.i.d. laws includes i.i.d. products over mixtures as well. To this end, let us define the
+mixture classes as below.
+Definition Let P be a set of distributions on Rd. For k ∈ N, we let Pk be the class of laws formed by k-fold
+mixtures of laws in P, and denote P∗ = �
+k∈N Pk as the class of laws formed by finite mixtures of laws in P.
+Note that P∗ is well defined since Pk form an increasing set. The second key result shown in §A.2 is
+Lemma 5. Let P ⊂ D be a m.a.c. set of laws on Rd. Then f-conv(P∞) ⊃ P∞
+∗ .
+The key observation underlying the above is already demonstrated in showing that f-conv(P∞) ⊃ P∞
+2 . To
+see this, fix any P, Q ∈ P, and α ∈ [0, 1]. We need to demonstrate a sequence of laws RT constructed via
+repeated fork-convex combinations that match the density process of R := (αP + (1 − α)Q)∞ for times up to
+T . This is realised as follows:
+R0 := P ∞, ST :=
+�
+RT −1 T −1,α
+−→ Q∞�
+, RT :=
+�
+ST
+T,0
+−→ P ∞�
+.
+In the above, ST matches the density process of R up to time T by mixing between RT −1 (whose tail behaves
+as P ∞) and Q∞ appropriately. RT then switches the tail of ST to behave as P ∞ to enable the recursion. This
+argument extends to P∞
+k
+for any arbitrary k by inducting over k (which is possible since a member of Pk is a
+mixture of a Pk−1 law and a P law). Since k is arbitrary, this immediately extends to P∗.
+Finally, we exploit the closure properties of fork-convex hulls under L1(Γ) to extend fork-convex hulls from
+product measures over a set to product measures over closures of that set.
+Lemma 6. Let P be a set of distributions on Rd that have densities. Then f-conv(� P) ⊃ � P, where P is
+the L1(Γ)-closure of P.
+The above lemma is a straightforward consequence of the closure properties as detailed in §A.2.
+10
+
+3.3
+Proof of the Absence of Nontrivial Test Martingales
+The previous section demonstrates that taking closed fork-convex hulls can yield significant expansion to
+product laws over sequences. This section exploits these properties to demonstrate the triviality of G∞-NSMs.
+The key observation underlying this is the following standard fact about the richness of Gaussian mixtures.
+Recall that P denotes the L1(Γ)-closure of P.
+Lemma 7. G∗ is L1(Γ)-dense in the set of all distributions with densities, i.e., G∗ = D.
+The L1(Leb)-denseness of mixtures of Gaussians in D is a classical fact; for instance see the work of
+Alspach and Sorenson [AS72] or Lo [Lo72]. More recently, a considerably more robust result was presented by
+Bacharoglu [Bac10], who shows that Gaussian mixtures are dense in nonnegative simple functions in both an
+L1 and an L∞ sense. This also suffices for our purposes since nonnegative simple functions are themselves L1-
+dense in nonnegative integrable functions. The L1(Γ)-denseness follows since Γ admits a uniformly bounded
+density with respect to the Lebesgue measure.
+We note that our argument extends to any such set, i.e., to any P such that P = D. The Gaussians serve
+as a convenient witness within L for which this property holds. With this in hand, we proceed as below.
+Proof of Theorem 1. Let {Mt} be a G∞-NSM. First observe by Lemma 5 and Proposition 2 that as a conse-
+quence, {Mt} is also a G∞
+∗ -NSM. Next, by Lemma 4 and Proposition 2, it is further a (� G∗)-NSM. Similarly,
+by Lemma 6 and Proposition 2, we conclude that {Mt} is also a (� G∗)-NSM. Finally, by Lemma 7, we
+conclude that {Mt} is a (� D)-NSM.1
+We now argue that � D is too rich to admit nontrivial NSMs. The argument is by contradiction—we
+assume that Mt > Mt−1 for some t with nontrivial probability, and use this to construct a law in � D that
+violates the NSM property. The argument repeatedly exploits the topological equivalence of (Rd)t and Rdt
+under the product and metric topologies respectively. We shall denote the Lebesgue measure in m dimensions
+as Lebm, and we note that the product Lebesgue measure on (Rd)t is identical to Lebdt, and use the latter to
+denote the former.
+Let us proceed with the argument. For a natural number t, define the event At := {Mt > Mt−1, Mt−1 < ∞},
+i.e. that {Mt} increases at time t. It suffices to argue that no matter the t, the mass of At is zero, since
+Γ(Mt−1 = ∞) must be zero due to integrability of Mt−1. For the sake of contractiction, assume Γ(At) > 0. For
+n ∈ N, define the approximations An
+t := {Mt ≥ Mt−1 + 1/n, Mt−1 ≤ n}. The An
+t form an increasing sequence
+of sets, and converge to {Mt > Mt−1, Mt−1 < ∞} = At.
+Now, since Γ(At) > 0 and At ∈ Ft, we conclude that Lebdt(At) > 0 due to the mutual absolute continutity of
+Gaussians and Lebesgue measures on Euclidean spaces. Without loss of generality, we may assume Lebdt(At) <
+∞ (since otherwise we may pass to a subset of At such that of positive and finite mass, using sigma-finiteness
+of the Lebesgue measure, and run the argument on this subset). Since An
+t ր At, we have by regularity of
+measure that Lebdt(An
+t ) → Lebdt(At), and in particular there exists an n such that Lebdt(An
+t ) ∈ (0, ∞). Fix
+such an n for the remainder of the argument.
+Recall that an open rectangle in Rm is a Cartesian product of open intervals, i.e.
+a set of the form
+×
+m
+i=1(ai, bi) for ai < bi. Similarly, we say that R is an open rectangle in (Rd)t if there exist open Rd-rectangles
+S1 . . . St such that R =×
+t
+s=1 Ss. The following statement is a consequence of basic topological and measure
+theoretic properties of Euclidean spaces, which we prove in §A.3.
+Lemma 8. Let E ⊂ (Rd)t be such that Lebdt(E) > 0. For every natural m ∈ N, there exists an open rectangle
+R in (Rd)t such that
+Lebdt(R) > 0
+and
+Lebdt(R ∩ E) ≥
+m
+m + 1Lebdt(R).
+Exploiting the above result, we may construct a sequence of rectangles in (Rd)t, {Rm}m∈N each of positive
+mass such that
+Lebdt(An
+t ∩ Rm)
+Lebdt(Rm)
+≥
+m
+m + 1.
+1We can also argue this more directly:
+observe that taking closed fork-convex hull is an idempotent operation, i.e.
+f-conv(f-conv(P)) = f-conv(P) (which follows from the facts that closed fork-convex hulls are fork-convex, and that closures
+of closed sets are invariant). Therefore, using the chain of Lemmata of §3.2, f-conv(G∞) ⊃ � G∗, and so {Mt} is a (� D)-NSM.
+11
+
+Now, since each Rm is a rectangle, there exists a law Dm ∈ � D such that the prefix restriction Dm|t =
+Unif(Rm). Indeed, if Rm =×
+t
+s=1 Sm
+s , then Dm = �{Dm
+s }, where Dm
+s = Unif(Sm
+s ) for s ≤ t, and Dm
+s = Γ for
+s > t. We claim that for large m, Dm witness a violation of the NSM property for {Mt}. We demonstrate this
+using the process {Nt} := {min(Mt, n + 1)}.
+Notice that if {Mt} is a P-NSM, then so is {Nt}, since
+E[Nt|Ft−1] ≤ min(E[Mt|Ft−1], E[n + 1|Ft−1]) = min(Mt−1, n + 1) = Nt−1,
+and the nonnegativity follows since both Mt and n + 1 are nonnegative. Further, since Mt−1 ≤ n on An
+t , it
+follows that Nt ≥ Nt−1 + 1/n on An
+t as well, since n + 1/n ≤ n + 1.
+Consequently, we have
+EDm[Nt] ≥ EDm
+�
+(Nt−1 + 1/n)1{Xt
+1 ∈ An
+t }
+�
++ 0
+= EDm
+�
+Nt−11{Xt
+1 ∈ An
+t }
+�
++ Dm(An
+t )
+n
+≥ EDm
+�
+Nt−11{Xt
+1 ∈ An
+t }
+�
++
+m
+n(m + 1),
+where the final inequality exploits the fact that at least a m/(m + 1) fraction of the mass of Rm lies in An
+t ,
+and we have used the nonnegativity of Nt.
+However, since Nt−1 is upper bounded by n + 1, we observe that
+0 ≤ EDm[Nt−11{Xt
+1 ∈ (An
+t )c}] ≤ (n + 1)Dm((An
+t )c) ≤ n + 1
+m + 1,
+and so
+EDm[Nt−11{Xt
+1 ∈ An
+t }] = EDm[Nt−1] − EDm[Nt−11{Xt
+1 ∈ (An
+t )c] ≥ EDm[Nt−1] −
+n + 1
+(m + 1).
+But now, we conclude that
+EDm[Nt] ≥ EDm[Nt−1] + (m/n) − (n + 1)
+m + 1
+.
+Choosing m > 3n2, and exploiting n ≥ 1, this implies that EDm[Nt] > EDm[Nt−1], thus contradicting the su-
+permartingale property of {Nt} under Dm (since supermartingales must have non-increasing mean sequences).
+We conclude that it cannot hold that Γ(At) > 0, i.e., Mt ≤ Mt−1 Γ-almost surely.
+But, since t is arbitrary, we immediately conclude that
+Γ(∃t ≥ 2 : Mt > Mt−1) ≤
+�
+t≥2
+Γ(Mt > Mt−1) = 0.
+The argument for NMs follows from this as well. If {Mt} is a � D-NM, then it is also an NSM, and thus
+almost surely does not increase. But this means that M1 − Mt ≥ 0 is also a nonnegative supermartingale, and
+therefore does not increase, which implies that Mt also does not decrease almost surely.
+Remark. It may be possible to develop a different argument that does not explicitly need to pass through
+the notion of fork-convex hulls. Perhaps one could directly work with the An
+s above, and replace Dm a by
+sufficiently skinny Gaussian Gm such that Gm(An
+s ∩ R) ≈ Gm(R) ≈ 1. However, there would still be sufficiently
+many technical details to iron out, so such an approach is not necessarily shorter or cleaner. More importantly
+however, our chosen path of development above leads to a richer characterisation of fork-convex hulls of
+i.i.d. processes with densities, and further directly illustrates the utility of such a characterisation. It thus
+deepens our understanding of the important geometric concept of fork-convexity.
+4
+The Sequential Universal Likelihood Ratio E-Process
+We begin by recalling the definition of e-processes from the introduction.
+12
+
+Definition An {Ft}-adapted process {Et} is said to be an e-process for a set of sequential laws P if
+sup
+P∈P
+sup
+τ E[Eτ] ≤ 1,
+where the second supremum is over all stopping times. Further, if for some n ≥ 1 it holds a.s. with respect to
+all P ∈ P that E1 = E2 = · · · = En−1 = 1, then we say that Et is an e-process for P started at time n.
+Next, we define the universal likelihood ratio (ULR) process [WRB20], which forms the main object of
+interest for this section.
+Definition Let E denote a sequence of estimators {Et}t≥0 such that each Et : (Rd)t → D. At any t, denote ˆqt =
+Et(X1, . . . , Xt). Finally, let ˆpt denote the log-concave maximum likelihood estimate over the data X1, . . . , Xt
+(which exists if t > d). The ULR process is the statistic
+Rt(Xt
+1; E ) := 1{t ≤ d} + 1{t > d}
+�
+d+1≤s≤t
+ˆqs−1(Xs)
+ˆpt(Xs) .
+We shall often suppress the dependence of Rt on Xt
+1 and E . The initial setting of Rt = 1 for t ≤ d is to
+account for the fact that log-concave MLEs are known to exist only if at least d + 1 samples are available.
+As discussed in the introduction, {Rt} constitutes an e-process due to the predictability of ˆqt−1 and the
+fact that they are probability densities. We formally state the validity of Rt as a proposition.
+Proposition 9. For any E , the process {Rt} is an e-process for L∞ started at time d + 1. Consequently,
+rejecting the null hypothesis when Rt ≥ 1/α results in an α-valid test for log-concavity.
+On exact MLEs. The measures ˆpt need not exactly maximise the likelihood ratio in the above. Indeed, if
+instead of the exact log-concave MLE ˆpt we instead an estimate ˜pt such that
+�
+s≤t
+log ˜pt(Xs) ≥ − log(1/ε) +
+�
+s≤t
+log ˆpt(Xs),
+then εRt ·� ˆpt(Xs)
+˜pt(Xs) is an e-process, and this can be thresholded at 1/α as before. This observation is pertinent
+since practical procedures for computing the log-concave MLE of a dataset are inexact, and only approximate
+the solution up to a (user-specified) additive gap in the log-likelihood objective, and require computation that
+scales polynomially with the inverse of this additive gap.
+For the remainder of this section, we shall equate laws P ∈ D1 with their density, denoted p.
+4.1
+Consistency of the ULR E-Process for Testing Log-Concavity
+Consistency of the ULR e-process depends strongly on the underlying estimator E . Indeed, as an extreme
+example, consider the case of ˆqt(Xt) = 1{Xt = X1}, for which the resulting Rt is a.s. 0 for any time t ≥ d + 1
+so long as the law P is continuous, and the test is thus powerless against such laws.
+It thus follows that the ULR e-process can only yield power against a set of laws determined by the
+estimator E . Concretely, we shall argue the same against the following set of ‘well estimable’ laws. Below, dH
+below denotes the Hellinger distance.
+Definition For a sequential estimator E , and a density p ∈ D1, define the prediction regret for a sequence
+{Xt} as
+ρt(E ; p) :=
+�
+s≤t
+log p(Xs) − log ˆqs−1(Xs).
+Further, let Lp denote the log-concave M-projection of p. We define the class of distributions that are well
+estimable by E with respect to log-concavity as
+Q(E ; c) :=
+�
+p ∈ D1 : P ∞
+�
+lim sup
+t→∞
+ρt(E ; p)
+td2
+H(p, Lp) ≤ c
+�
+= 1
+�
+.
+13
+
+The main result of this section is that the ULR-based test is powerful against the above well-estimable
+laws, which is shown later in this section.
+Theorem 10. There exists a constant c > 1/25 such that if p ∈ Q(E ; c) \ L, then P ∞(Rt → ∞) = 1.
+Consequently, the ULR e-process yields a consistent test against i.i.d. draws from any distribution in Q(E ; c).
+The well-estimability condition above essentially requires that the distribution can be estimated well in a
+log-loss sense. For i.i.d. distributions, one expects that for reasonable E , the estimates ˆqt converge to some ˆq,
+and thus the regret grows for large t as ρt ≈ tKL(p∥ˆq) (which could grow sublinearly in t if KL(p∥ˆqt) → 0, but
+the latter convergence is not required). The class Q thus roughly consists of distributions can be estimated well
+in KL divergence. Such estimation can be a challenging task in complete generality, since the KL divergence is
+quite sensitive to mismatch in the tails of distributions. However, under mild restrictions such as compactness
+of support and smoothness, such estimability is quite forthcoming. Indeed, we give the following statement to
+illustrate this point. This is proved in §B.3.
+Corollary 11. Let DBox,Lip,B denote the set of 1-Lipschitz densities supported on the unit box [−1, 1]d and
+bounded between [1/B, B]. There exists a sequence of sieve maximum likelihood estimators E such that for
+every c > 0, DBox,Lip,B ⊂ Q(E ; c), i.e., the ULR e-process yields a consistent test against i.i.d. draws from
+such distributions.
+It is further interesting that the consistency of the test does not require that the regret ρt/t → 0, and
+only that it gets small enough relative to the Hellinger distance between p and its log-concave M-projection
+Lp. This signals that deviations from log-concavity may be detected far before the underlying law can be
+estimated, which is quite favourable theoretically, although its practical effects depend significantly on how
+large a c can be taken in Theorem 10.
+Proof of Theorem 10. We begin by defining σt(p) = �
+s≤t log p(Xs) − log ˆpt(Xs). Observe that
+log Rt = σt(p) − ρt(E ; P).
+Further, by assumption, we have that p ∈ Q(E , c) for some c, and thus for any ζ > 0, we have that
+ρt ≤ (1 + ζ)ctd2
+H(p, Lp),
+for large enough t. Consequently, to show that Rt → ∞, it suffices to show that P ∞ almost surely,
+lim inf
+t→∞
+σt
+td2
+H(p, Lp) ≥ (1 + 2ζ)c.
+(3)
+It is at this point that the following lemma is useful, the proof of which is left to §B.1.
+Lemma 12. For any p ∈ D1, it holds that
+P ∞
+�
+lim inf
+t→∞
+σt(P)
+td2
+H(P, LP ) ≥ 1
+25
+�
+= 1.
+The claim (3) thus follows so long as (1 + 2ζ)c ≤ 1/25, and since ζ > 0 can be taken arbitrarily small,
+this allows us to take any c < 1/25. We note that the constants in this argument are loose, and informal
+calculations suggest that it may be possible to improve c up to about 1/6.
+The proof of Lemma 12 relies on strong convergence properties of the log-concave MLE ˆpt to the log-concave
+M-projection Lp. Recall that for a pair of functions u ≤ v, a bracket [u, v] is the set of all functions that lie
+between u and v everywhere. By exploiting a characterisation of the convergence properties of log-concave
+MLEs due to Cule and Samworth [CS10], Dunn et al. [DGWR21, Lem. 1] show that there is a small bracket
+that is well separated from P such that ˆpt eventually lies in this bracket. Conditioning on this event, we
+then exploit a classical result of Wong and Shen [WS95] which show linear growth of σt(p) with condiitonal
+probability at least 1−exp (−Ω(t)) , at which point the lemma follows by Borel-Cantelli. As mentioned before,
+see §B.1 for the full proof of Lemma 12.
+14
+
+4.2
+Power of the ULR E-Process for Testing Log-Concavity
+The argument underlying Theorem 10 is also amenable to deriving rates, under further restrictions on the
+underlying law P. As in the previous section, we argue this using the decomposition log Rt = σt(p) − ρt(E ; p).
+4.2.1
+Challenges, and Context from the Theory of Log-Concave MLEs
+With the above approach, the argument breaks into two parts. Frstly, we assume that we use a good enough
+estimator E so that ρt is not too large with high probability. Such an assumption is necessary for the approach
+we take, although in principle the test can be analysed using a different decomposition, in which case this
+assumption may perhaps be weakened. In any case, we observe that for concrete alternate hypotheses such as
+laws with Lipschitz densities supported on the unit hypercube, ρt can indeed be appropriately controlled. It
+is worth noting, however, that the resulting rate bounds are strongly driven by the behaviour of ρt, and thus
+the estimator being considered, which limits the power of the results to follow.
+The second part of the argument requires us to show that σt is large, i.e., to argue that the log-concave
+MLE cannot represent the underlying law very well when it is not log-concave. While a natural statement,
+arguing this is challenging because this requires us to understand the behaviour of log-concave MLEs ‘off-the-
+model,’ i.e., when the data is not drawn from a log-concave distribution itself. With the notable exception of
+Barber and Samworth [BS21], this task has not been undertaken in the literature, with most works focusing
+on on-the-model minimax rate bounds [KS16; KDR19; Han21; CDSS18]. Let us consider this in some detail.
+Tight analysis of the on-the-model log-concave estimation problem fundamentally relies on a subtle re-
+duction of the rates of log-concave MLEs to the problem of controlling deviations of empirical processes over
+convex sets, i.e., to that of controlling supC |P(C) − Pt(C)| under data drawn from P, where Pt is the empir-
+ical law, and the supremum is over convex sets in a bounded domain [CDSS18]. Using this observation and a
+refined study of these deviations, Kur et al. [KDR19] recently showed tight on-the-model estimation rates of
+the form dH(ˆpt, p) = O(n−1/(d+1)) when p ∈ L and d ≥ 3 (Han showed similar results, along with extensions
+to s-concave densities [Han21]). While significant elements of this study can be extended to analysing off-
+the-model behavior, the analysis ultimately cannot be applied to our situation. The gap arises because their
+argument only upper bounds the quantity
+θt := EX∼P [log Lp(X)] − EX∼P [log ˜pt(X)],
+where for a small constant c, ˜pt ∝ max(c, pt) is a slight modification the log-concave MLE. When p = Lp, this
+object is a KL-divergence, and so is lower bounded. However, when p ̸= Lp (that is, P is not log-concave),
+this quantity is may well be negative.
+Notice that this is a problem for us precisely because E[σt]/t ≈
+θt + EX∼p[log p(X) − log Lp(X)]. When p ̸∈ L, the second term can indeed be shown to be large, but the lack
+of a lower bound on the first term limits the applicability of such results. We also note that other aspects
+of the argument, which are relatively simple in on-the-model analysis (for instance, arguing that the mass p
+places on sets of the form {Lp(x) < γ} is small), are also rendered inoperative in off-the-model analysis.
+Of course, we can in principle exploit the results of Barber and Samworth instead. However, these re-
+sults give quite poor rates. Roughly speaking, Theorem 5 of their paper [BS21] shows that off-the-model,
+dH(ˆpt, Lp) ≲ t−1/4d, and thus any analysis that exploits this result cannot hope to show that σt is large for
+t ≪ dH(p, L)−4d. This power of 4d arises since the analysis of [BS21] passes through a reduction to convergence
+of empirical laws in Wasserstein distance (which gives the relatively benign factor of d), and further suffers a
+1/4th power slowdown relative to this convergence (which is both unavoidable, and leads to a 4d exponent).
+Our analysis sidesteps these issues by controlling the growth of σt on the basis of bracketing entropy (see
+§B.1) bounds for the class of bounded log-concave laws on compact supports. Our bound below holds for all
+d but appears to be new for d ≥ 4. Indeed, we show the following statement.
+Lemma 13. Let Ld,B denote the set of laws with log-concave densities that supported on [−1, 1]d and uniformly
+upper bounded by a constant B. There exists a constant Cd dependending only on the dimension such that
+H[](Ld,B, ζ) = Cd �Θ
+�
+(B/ζ)max(d/2,d−1)�
+,
+where the �Θ hides terms that scale polylogarithmically with ζ or B.
+15
+
+We note that the lower bounds on the bracketing entropy implicit in the statement of Lemma 13 were
+already shown by Kim & Samworth [KS16, Thm. 8], who further also showed the corresponding upper bounds
+for d ≤ 3. While not the central point of the paper, we develop upper bounds for the same when d ≥ 4.
+There are two salient technical points regarding the bound above. Firstly, observe that for d ≥ 4, the entropic
+bounds lie in the non-Donsker regime, i.e., when the Dudley integral
+� ε
+0
+�
+H[](Ld,B, ζdζ does not converge due
+to a blow-up near ζ = 0, which typically (but not always) represents a slowdown in the convergence rates that
+can be shown via entropy integrals. Secondly, for d ≥ 3, the bound grows as ζ−(d−1) rather than as ζ−d/2.
+The latter quantity is pertinent it is close to the growth rate of the bracketing entropy of convex sets which
+is ζ−(d−1)/2; see §B.2.3. This fact underlies the power of the previously discussed reduction of the analysis
+of log-concave MLE rates to control on the deviations of empirical processes over convex sets, which admit a
+slower entropy growth.
+Lemma 13 is proved in §B.2.3. The growth bounds of this result ensure that for t ≳ dH(p, L)2(d−1), σt is
+linearly large in t, even when the underlying law is not log-concave. This 2(d−1) exponent should be compared
+to the aforementioned 4dth power scaling that one expects to emerge from using the Wasserstein continuity
+based approach discussed above. Of course, the dependence on d could potentially be improved even further.
+For instance, if the on-the-model analysis can indeed be extended to off-the-model, it is plausible to expect
+dependence of the form d + 1 instead of 2d − 2. However, this remains a challenging problem for future work.
+It is worth noting that while the techniques for the bounds in Lemma 13 exist in the literature, the
+bounds themselves appear to not have explicitly been observed. We believe that this might be because it was
+previously observed that due to existing lower bounds on this entropy, the resulting growth rate bounds that
+emerged from entropic considerations could not be optimal for the rate analysis of the log-concave MLE, at
+least in on-the-model settings. It should also be noted that the bounds above are explicitly for compactly
+supported log-concave laws (which is a restriction, but a relatively mild one, due to the exponentially decaying
+tail enjoyed by all log-concave densities). Further note that that the brackets we construct for this setting
+are ‘improper’, i.e. the bracketing functions are themselves not log-concave, which may limit utility in direct
+analysis of the difference between ˆpt and Lp, but is good enough when studying the behaviour of σt.
+4.2.2
+Bounding Typical Rejection Times for the ULR Test
+As discussed previously, our analysis of σt passes through a bracketing entropy bound for bounded, compactly
+supported log-concave laws. For such bounds to be effective, we need to ensure that the log-concave MLE ˆpt
+itself is bounded. This is enabled by the quantity ∆P , defined for a law P as
+∆P :=
+min
+v:∥v∥=1 EP [|⟨v, X − EP [X]⟩|],
+which was identified by Barber and Samworth [BS21] as a means to lower bound the covariance of the log-
+concave projection of P, which in turn can be exploited to upper bound the supremum of Lp and (indirectly)
+ˆpt. Observe that ∆P roughly corresponds to the minimum eigenvalue of the covariance matrix of P — indeed,
+it is best seen as a robust version of the same.
+With this in hand, we are ready to state our main result, the proof of which is the subject of §B.2.2. Recall
+that dH(p, L) = infq∈L dH(p, q) and τα := inf{t : Rt ≥ 1/α} is the rejection time.
+Theorem 14. Suppose p is supported on [−1, 1]d, and let πt → 0 be a sequence such that for every t,
+P ∞
+� ρt(E , p)
+td2
+H(p, L) ≥ 1
+25
+�
+≤ πt.
+Then there exists a constant c ∈ [1/600, 1] and a natural number T0 such that for any t ≥ T0 + log(1/α)
+cd2
+H(p,L),
+P ∞ (τα > t) ≤ πt + 1
+c exp
+�
+−ctd2
+H(p, L)
+�
++ 1
+c exp
+�
+−ct∆2
+P /d2�
+,
+and
+T0 = Cd · �O
+�
+∆− max(d2/2,d2−d)
+P
+dH(p, L)− max((d+4)/2,2(d−1)) + d2∆−2
+P
+�
+,
+where Cd depends only on d, and the �O hides terms depending polylogarithmically on dH(p, L), ∆P .
+16
+
+Observe that from the statement above we may conclude that the average rejection time is bounded as
+E[τα] =
+�
+t
+P ∞(τα > t) ≤
+�
+t
+πt + O(T0).
+Here, the first term is driven by the predictability of p using the estimators E , while the second term is driven
+by our analysis of the noise scale of log-concave density estimation in off-the-model scenarios.
+In typical situations, the former of these terms will dominate the resulting bounds, since typical alternate
+classes will be much larger than the class of log-concave distributions. For instance, using results on the
+estimation of uniformly lower-bounded Lipschitz densities [WS95, e.g.], we show the following result about
+the set DBox,Lip,B introduced in Corollary 11.
+Corollary 15. For any constant B > 0, there exists a sequence of sieve maximum likelihood estimators E such
+that if p ∈ DBox,Lip,B, the ULRT rejection time τα is bounded in expectation as E[τα] = �O(dH(p, L)−2(d+3)).
+The proof is in §B.3. We remark that the above rates adapt to the extent to which the underlying law
+p violates log-concavity in the sense that the time-scales of rejection are driven by dH(p, L). Indeed, this
+represents an important advantage of sequential tests as opposed to batched tests, in that validity is retained,
+and detection is guaranteed at an adapted time-scale.
+On tightness. We note that the exponents of Theorem 14, and in particular, Corollary 15 are likely loose
+for the problem of testing log-concavity. This is an artefact of the analysis; for instance, the slow rate in
+Corollary 15 is largely determined by the rate requirements for estimating Lipschitz laws on the unit box,
+which arises due to the πt terms present in Theorem 14. It is possible that this aspect can be improved, since
+nothing neccessitates that we use an estimator that captures the underlying density p well.
+Indeed, instead of analysing the prediction regret with respect to p itself, we could decompose R = σt(q) −
+ρt(E ; q) for some other law q, perhaps lying in a smaller class of densities Q than those possible for p. As long
+as (i) E does as good a job at prediction under the log-loss as any law in Q, and (ii) no matter what p ̸∈ L
+is, there is a law in Q that is ‘closer to’ p than any law in L, a similar analysis should be possible, although
+this requires possibly subtle off-model control on the behaviour of E , as well as a careful choice of Q itself
+to control the relative values of distances such as dH(p, Q) and dH(p, L). One such approach which appears
+promising for log-concave laws is to exploit s-concave densities to play the role of Q, which are particularly
+attractive since they form a rich extension of the class of log-concave laws, but nevertheless enjoy identical
+minimax MLE convergence rates as them [HW16; Han21].
+5
+Algorithmic Proposal, and Simulation Study
+We now proceed to algorithmically describe the ULR e-process based test for log-concavity, and investigate
+the behaviour of a concrete implementation of the same on a simple parametric family.
+5.1
+Computational Aspects, Batching, and a Concrete Testing Algorithm
+Under specification of the sequential estimators E , and a method for fitting the log-concave MLE, the statistic
+Rt is explicitly computable, and thus naturally leads to implementations. While the e-process is powerful
+against wide classes of alternatives, its implementation suffers from a fundamental computational issue, that
+arises due to the recomputation of ˆqt and ˆpt in each round. This cost grows superlinearly with t since since the
+entire denominator � ˆpt(Xs) must be evaluated on the entirety of the stream, and the cost of estimating this
+ˆpt is itself superlinear in the number of samples t. A second issue arises upon increasing the data dimensions d,
+since computational costs of estimating ˆpt grow quite fast with this. Even though polynomial in d algorithms
+exist for computing the log-concave MLE [Axe+19], the fastest available method for this is typically hundreds
+of times slower when processing ∼ 100 points in even the modest d = 5 when compared to the time needed
+to process the same sized dataset for d = 1 [RS19]. We address this issue by exploiting batching to reduce
+the computational load, which makes computations viable in the moderate d ≤ 4. The idea is to wait to
+17
+
+accumulate I > 1 fresh samples before recomputing Rt, rather than updating it at every round.2
+Let us point out that such batched updates still retain the e-process property, and thus validity, as long
+as the ˆqt−1 remain nonanticipating over the entire batch. Concretely, we may set a schedule, captured by an
+increasing sequence of times T = {tk}, and evaluate the statistic
+Rt(T ) = Rt−1(T )1{t ̸∈ T } +
+�
+1{t ∈ T }
+�
+j≤k(t)
+tj
+�
+s=tj−1
+ˆqtj−1(Xs)
+ˆptk(Xs) ,
+where k(t) = max{k : tk ≤ t}. In words, the schedule divides streams into a sequence of batches of size
+tk − tk−1, and each time a new batch is accumulated, we evaluate a new estimate ˆq on the previous batches,
+and re-evaluate the log-concave MLE on the entirety of the data seen. This process continues to be dominated
+by a batched version of Ft(P), which retains the martingale property under P ∞, thus yielding validity. The
+simplest viable schedule is to set tk = kI for a constant ‘batching interval’ I. This effectively boils down to
+testing the log-concavity of p⊗I, which is valid since tensor products of log-concave laws remain log-concave.
+Notice that such batching may result in a reduction in power. For instance, rejection can only occur at the
+time tk, and further the statistic may be deflated because data points with a large signal may be ‘washed-out’
+due to milder behaviour across the remainder of the batch. Nevertheless, we find in simulation studies that
+this drop in power is nominal, and comes at the cost of a significant improvement in runtime.
+With this in hand, we can provide an explicit algorithmic description of our test below.
+Algorithm 1 Log-Concave Universal Likelihood Ratio Test
+1: Input: Batching schedule {tk}∞
+k=1 with t1 ≥ d + 1, estimator E , level α.
+2: Initialise: Rt ← 1 for t ≤ t1, K ← 1, N1 = 1, t ← 1.
+3: while Rt < 1/α do
+4:
+if t = tK then
+5:
+ˆq = E (Xt−tK−1
+1
+).
+6:
+Nt ← Nt−1 · �tK
+s=tK−1+1 ˆq(Xs).
+7:
+ˆp ← L (XtK
+1 ).
+8:
+Rt ← Nt ·
+��tK
+s=1 ˆp(Xs)
+�−1
+.
+9:
+K ← K + 1.
+10:
+else
+11:
+Rt ← Rt−1.
+12:
+Nt ← Nt−1.
+13:
+t ← t + 1.
+5.2
+Evaluating the ULR E-Process Test
+We investigate the behaviour of the test of Algorithm 1 on the following simple test-bed family of laws, where
+ed = 1d/
+√
+d is the unit vector along the all-ones direction in Rd,.
+p(x; µ, d) :=
+1
+2(2π)d/2
+�
+exp
+�
+−∥x − µ
+2 ed∥2/2
+�
++ exp
+�
+−∥x + µ
+2 ed∥2/2
+��
+,
+i.e., balanced two component Gaussian mixture laws with means ± µ
+2 ed and identity covariance. The norm
+of the mean-difference is precisely µ, which we assume without loss of generality to be nonnegative. A small
+modification of this family of laws was also used as a test-bed for the non-sequential test proposed by Dunn
+et al. [DGWR21].
+These laws are extremely convenient for proof-of-concept investigation of tests of log-concavity. Indeed,
+observe that up to a rotation, the d-dimensional law is a tensorisation of the one-dimensional law with a
+2Note of course, that for our simulation study, the repetition of simulations required to study power and size mean that we
+only implement our fully nonparametric test for up to d = 4. Nevertheless, even this is reasonable to run for d = 6, wherein a
+single run over a horizon of 100 steps takes about 20s.
+18
+
+log-concave law (specifically a standard Gaussian in d − 1-dimensions). Since log-concavity properties are
+invariant under rotations, and since the log-concave M-projection of product laws is a product of the marginal
+log-concave M-projections [SW14], this gives a very simple characterisation of the log-concavity properties of
+this law. Concretely, the distance from log-concavity is purely a function of the norm of the mean-difference
+µ, and p(x; µ, d) is log-concave if and only if µ > 2. These laws thus give us a simple way to check both the
+size and power of the test statistics, as well as study the effect of increase in dimension on the power.
+Finally, we give details of the simulations. All data reported is a mean over 100 runs of each experiment.
+All simulations are run up to 100 time steps, which is mainly for computational practicality. Note thus that
+our size estimates are systematically lower than the true size (with infinte horizon). We run the case of d = 1,
+which is computationally the cheapest, over the longer horizon of 500 time steps to illustrate that not much
+changes in this case, at least as regards the empirical validity of our test. For d = 1, 2, 3, the tests are batched
+at an interval of I = 20, while for computational practicality the test is batched at I = 25 for d = 4. These
+are significant fractions of the time horizon studied, but do not significantly lower power, at least for d = 1,
+as demonstrated by explicit simulation.
+All code is implemented in R. The nonparametric estimator used for E is the kernel density estimate
+as implemented in the ks package [CD18], and the log-concave MLE used is either from the logConDens
+package [DR11] for d = 1 or the fmlcd package (d = 2, 3, 4) [RS19]. We note that the latter is not guaranteed
+to return the log-concave MLE since it optimises a non-convex approximation to the program defining the
+same. However, we find that compared to alternatives like the logConcDEAD package [CSS10], the fmlcd
+implementation retains similar validity and power, but runs significantly faster. We also investigate using
+parametric Gaussian mixture model fits to illustrate the effect of inefficiency in E on power, for which we use
+the EM algorithm as implemented in the mclust package [SFMR16]. All simulations were executed on an
+AMD Ryzen 5650U processor, a medium range CPU for a laptop computer.
+5.2.1
+Fully Nonparametric tests
+Figure 2 shows the behaviour of our instantiation of the algorithm with the fully nonparametric approach of
+using kernel density estimators as E over p(·; µ, d) for d = 1, 2 as µ is varied, run at the size α = 0.1, with
+I = 20 for d ∈ {1, 2, 3} and I = 25 for d = 4. We plot five traces which record the fraction of runs out of 100
+independent runs that the test rejected the null hypothesis at times smaller than 20, 40, 60, 80, and 100, where
+100 was the horizon over which the test was run.
+There are three major observations. Firstly, we observe that the test shows excellent validity. Indeed, the
+null hypothesis holds true for µ ≤ 2, and the test does not reject more than a 0.02 fraction of the time in
+either case in this scenario. Secondly, we observe that at least for sufficiently large µ, all of the tests do reject
+within 100 steps. Finally, we notice that the power sharply drops as d increases. To concretely discuss this,
+let µ∗(d) be defined as the smallest value of µ for which PX∼p(·;µ,d)(τ0.1 ≤ 100) = 0.9. The plots in figure 2
+give us estimates of µ∗(d), which increase sharply with d—from about 6 in d = 1 to over 1000 in d = 4.3
+This reduction in power is perhaps expected, given the considerable deterioration in the nonparametric
+estimation rates with d. Nevertheless, we may question how much of the above decay in power is driven by
+the inefficiencies in fitting log-concave MLEs, and how much accrues due to the inefficiency of kernel density
+estimation. We investigate this effect in §5.2.2 by studying Oracle tests.
+Longer Run for d = 1.
+To show that the validity persists over longer time horizons, we implement the
+fully nonparametric method over 500 time steps for d = 1, using I = 50. Observe in Figure 3 that rejection
+under the null µ ≤ 2 is well controlled even at this increased timescale, while rejection rates steadily improve
+as the horizon grows, although the improvement is somewhat marginal over the horizon of 500 versus 200.
+3With pilot simulations in d = 5, we observe that µ∗(5) ≈ 1500. We note that these simulations were already too costly, in
+terms of time, to implement completely for d = 5, due both to the increased costs of fitting MLEs in higher dimensions, and due
+to the fact that as rejection rates decrease with dimension, more runs need to be executed over the whole horizon, which extends
+the total cost of the experiment. We hope to implement the method on larger computational resources for such moderate ds.
+19
+
+Figure 2: Performance of the fully nonparametric test. Empirical rejection rates (over 100 simulations)
+at α = 0.1 versus the mean difference µ for fully nonparametric test implementations over four cases: d =
+1, I = 20 (top left); d = 2, I = 20 (top right); d = 3, I = 20 (bottom left) ; d = 4, I = 25 (bottom right). The
+thin horizontal line plots the level α = 0.1, and the vertical line marks µ = 2 since p(·; µ, d) ∈ L ⇐⇒ µ ≤ 2.
+Observe the strong validity properties in all plots, as well as the deterioration of power in higher dimensions,
+as signalled by the sharp increases in the scales of the X-axis.
+Figure 3: Performance of the fully nonparamet-
+ric test over long horizons. Empirical rejection
+rates (over 100 simulations) in the setting of Figure 2
+for d = 1, ran over a horizon of length 500 with
+I = 50. Observe that the validity persists over this
+longer horizon, and that power improves for µ > 2.
+Figure 4: Effect of I on the fully nonparametric
+test. Empirical rejection rates (over 100 simulations)
+in the setting of Figure 2 for d = 1 and with varying
+I ∈ {1, 10, 20, 50}. Observe that the rejection rates
+for I = 10, 20 are roughly the same as for I = 1,
+while I = 50 suffers large losses.
+20
+
+Figure 5: Performance of the partial oracle test. Empirical rejection rates (over 100 simulations) at
+α = 0.1 versus the mean difference µ for the partial oracle test implementations over four cases: d = 1, I = 20
+(top left); d = 2, I = 20 (top right); d = 3, I = 20 (bottom left) ; d = 4, I = 25 (bottom right). Observe that
+in each plot, the power improves starkly relative to the fully nonparametric test (Figure 2), as indicated by a
+strong contraction of the scale of the X-axis, especially in higher dimensions.
+Effect of Batching Interval.
+As seen in Figure 4, batch sizes of I = 10 and 20 have a mild effect on the
+rejection rates under alternate setting (µ ≥ 2) when compared to the direct I = 1. Interestingly, note that
+I = 20 does somewhat better than I = 10 in the setting of moderate µ (the range 4 − 6), and slightly loses
+power for larger intermediate µ (the range 6 − 8). In turn, the no batching setting, i.e., I = 1, is observed to
+suffer deterioration in its size (µ < 2), although this remains at an acceptable level.
+The large batch size I = 50 suffers the same validity issues as I = 1, but does even better than it for small
+but non-null values of µ (2-5). Power considerably deteriorates for larger µ (5-10). While it is unclear how
+much of this is an artefact of the fact that the length of the horizon is only 100, and how much is directly
+due to the larger batching interval, the fact that I = 10, 20 perform well suggests that so long as the batching
+interval is a relatively small fraction of the horizon length, the loss in power is not too bad.
+5.2.2
+Oracle Tests, and the Effect of the Quality of E
+Oracle Tests.
+To probe the effect of the lossiness of the kernel density estimate on the power of the fully
+nonparametric test, we run ‘partial-oracle’ and ‘full-oracle’ oracle tests, which adjust E to exploit concrete
+information about the underlying laws p(·; µ, d). In the partial-oracle, we adjust E to estimate a two-component
+Gaussian mixture model instead of a kernel density estimate, and in the full-oracle case, we directly set
+ˆqt−1(·) = p(·; µ, d), i.e., we exactly evaluate the density.
+We expect that under data drawn from p(·; µ, d), these tests are more powerful than the fully nonparametric
+tests discussed above, since the regret ρt(E ; p) would reduce in the case of the partial oracle due to a reduced
+21
+
+Figure 6: Perforamance of the full oracle test. Empirical rejection rates (over 100 simulations) at α = 0.1
+versus the mean difference µ for the full oracle test implementations over four cases: d = 1, I = 20 (top left);
+d = 2, I = 20 (top right); d = 3, I = 20 (bottom left) ; d = 4, I = 25 (bottom right). Observe the sharp
+improvement in power compared to Figure 2, especially in high dimensions, as indicated by a strong contraction
+in the scale of the X-axis. Observe also the improvement in power compared to Figure 5, in that the curves
+reach high power at about half the µ that is needed for the partial oracle test.
+complexity of the estimation class; and, of course, would reduce exactly to 0 in the case of the full oracle. In
+either case, this effectively serves to increase Rt. These oracle tests thus let us probe the extent of the loss in
+power at a fixed µ (and thus a fixed distance from log-concavity) that arise purely due to the decay in rate of
+convergence of the log-concave MLE. In particular, the full oracle test captures exactly this effect, while the
+partial oracle test approaches this in a soft way. Figure 5 shows the performance of the partial oracle tests,
+and Figure 6 shows the same for the full oracle test for d ∈ {1, 2, 3, 4}.
+Comparing Figures 2 and 5, we see that for using the partial oracle yields a marked increase in power,
+at least for d > 1. This is evident in d = 2 by observing that the purple lines (overall rejection rate within
+100 times steps) rises higher and is nonzero at smaller values of µ, as well as observing that the typical
+rejection time decreases substantially (for instance, rejection never happened below time step 60 in the fully
+nonparametric case, but is quite prevalent at higher µs under the partial oracle). In d = 3, 4 the effect is
+much starker - notice that the scale of the plot completely changes, from order of hundreds to tens in d = 3.
+This suggest that using the parametric mixture of Gaussians estimate offers strong improvements over the
+nonparametric KDE estimate due to the reduced variance scale of this estimator.
+The above effect is seen even more starkly in the case of the fully oracle test, where each of the rejection
+rate curves is further improved (Figure 6). For instance, our estimate of µ∗(d) (the smallest µ such that
+Pp(·;µ,d)(τ0.1 ≤ 100) = 0.9)) is about halved for the full oracle case when compared to the partial oracle (and
+improved manifold relative to the fully nonparametric test).
+22
+
+The Quality of E has a Strong Effect.
+These observations from the oracle tests indicate that the quality
+of the estimate offered by E is very important in driving the overall power of the test. In these oracle examples,
+the quality improved by reducing the variance scale of the estimator, whilst keeping the bias at 0 (since the
+law p(·; µ, d) is representible by each of the estimator outputs).
+Of course, in practice we cannot always hope to reduce the variance scale of our estimates whilst keeping
+the bias zero. Nevertheless, there is a tradeoff between the two implicit here. Indeed, as we discussed briefly in
+§4, it is possible to use a biased E in the test, i.e. one that does not strictly estimate p, so long as the output of
+E does a better job of representing p than the log-concave MLE. The strong dependence of the testing power
+on E indicates the critical need to investigate this design freedom, and to study how the trade-off between
+the variance, in terms of the convergence rates of ˆqt, and the bias, i.e., the distance of lim ˆqt from p, should
+be balanced to optimise the testing power.
+6
+Discussion
+Our work has shown that the sequential testing of log-concavity throws interesting challenges, in that the
+prevalent paradigm of test martingales cannot be fruitfully applied to this practically relevant setting. In the
+process of doing this, we developed a characterisation of the closed fork-convex hulls of independent sequential
+laws on a continuous space, thus contributing to the theory of this new tool that characterises the nonnegative
+supermartingale property. We then showed that the universal likelihood e-process instead does yield powerful
+tests for log-concavity. In particular, we demonstrated that these tests are consistent against large classes of
+nonparametric alternate laws, and further admit nontrivial rates, and made contributions to the off-the-model
+analysis of the convergence of log-concave MLEs, as well as the general theory of the power analysis of universal
+tests in order to do so. These properties are validated by running the test over a simple parametric family
+of laws, which further demonstrates the critical role of the sequential estimator E in the power of the test.
+Taking a broad view, the above can also be seen as a contribution to the emerging literature on e-processes,
+and in particular as additional evidence for the case that the study of sequential testing at large must exploit
+this powerful yet simple tool.
+A number of directions, both theoretical and methodological remain open in this interesting subject, a few
+of which we discuss below.
+Regarding fork-convexity, our characterisation in §3 and §A of the closed fork-convex hulls of i.i.d. Gaus-
+sians can possibly be further enriched, and it would be very interesting to understand precisely which laws lie
+in this set. Additionally, notice that sequentially testing the Gaussiantiy of an i.i.d. process itself is a basic
+problem that again cannot be tested using martingales (at least with respect to the natural filtration of the
+data). Construction and analysis of such sequential Gaussianity tests is a natural and interesting direction.
+Of course universal inference is again a natural approach for this class, but it may be possible to take ad-
+vantage of translation and rotation invariance of the null hypothesis (all Gaussians) using methods developed
+in [PLHG22].
+Regarding the ULR e-process based test for log-concavity, §5 shows that the power of the fully nonpara-
+metric test can be quite limited particularly as the data dimension increases. This observation was also made
+in the non-sequential setting by [DGWR21], who proposed using random one-dimensional projections as an
+interesting method to ameliorate this. In this test, rather than computing the full d-dimensional kernel and
+log-concave estimates, one projects the data onto many one-dimensional subspaces, and averages the e-values
+(nonnegative test statistics with expectation at most one under the null) that result from a one-dimensional
+test carried on each of these projected datasets. This approach not only has computational benefits due to
+the speed of one-dimensional density estimation methods, but also shows statistical benefits in the scenario
+of §5, in that the decay of power is considerably limited with dimension. Such projected tests are of course
+possible in the sequential setting as well, and are a natural next step to investigate, both methodologically
+and in terms of their theoretical properties.
+On a broader scale, both the theoretical bounds and the simulations illustrate the critical role that the
+quality of the estimator E plays, both specifically in the power of the test for log-concavity, but also more
+generally in the use of the universal likelihood ratio e-process. With this in mind, and recalling the implicit
+‘bias-variance’ tradeoff in E as discussed in §4 and §5, investigating the choice of E relative to the null class
+23
+
+is an interesting question both in terms of practical methodological concerns, as well as theoretical concerns
+studying the power of e-process based tests.
+Acknowledgments
+The authors thank Martin Larsson for insightful discussions on fork convexity, and Robin Dunn, for an
+implementation for a batched universal test for log-concavity that formed the backbone of the code underlying
+our simulations. A. Rinaldo and A.G. were supported in part by the NSF grant DMS-EPSRC 2015489.
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+26
+
+A
+Proof of Triviality and Properties of Fork-Convex Hulls
+This appendix is devoted to showing the structural lemmata regarding fork-convex hulls, and discussing
+technical aspects of our arguments.
+A.1
+Details on the Local L1(Γ) Closure
+Let us begin by explicitly detailing the notion of convergence implicit in closed fork-convex combinations.
+Recall that the f-conv(P) is the closure of of f-conv(P) with respect to L1(Γ)-convergence of likelihood
+ratio processes at every fixed time t. Let us unpack this statement in simple terms. Let Pn be some sequence
+in f-conv(P) of density ratio Zn
+t := ZPn
+t . We say that Pn → P if for every t, it holds that Zn
+t → Zt in L1(Γ).
+Since Zt and Zn
+t are Ft measurable objects, this convergence is simply in L1(Γ|t). Stating that the convergence
+needs to happen at every fixed time t means that this convergence need not be uniform in t: it is fine for Zn
+100
+to converge more slowly than Zn
+1 , for instance. This notion of convergence may be metrised by
+∆(P, Q) :=
+�
+t∈N
+2−t∥ZP
+t − ZQ
+t ∥L1(Γ).
+We note that ∆ is bounded, since
+∥ZP
+t − ZQ
+t ∥L1(Γ) =
+� ���P|t(dxt
+1) − Q|t(dxt
+1)
+��� ≤
+�
+P|t(dxt
+1) +
+�
+Q|t(dxt
+1) = 2.
+With this in hand, we first show the following auxiliary claim that is repeatedly used.
+Lemma 16. Let P be a set of sequential laws, and let R be any sequential law. Suppose there exists a sequence
+of sequential laws {RT} such that each RT ∈ f-conv(P), and for all t ≤ T, ZRT
+t
+= ZR
+t . Then R ∈ f-conv(P).
+Proof. We claim that RT → R. Indeed, since ZRT
+t
+= ZR
+t for all t ≤ T,
+∆(RT , R) ≤
+�
+t>T
+2−t · 2 = 2−(T −1).
+Thus, limT →∞ ∆(RT , R) = 0, meaning RT → R. Since the closed fork-convex hull of P includes such limits by
+definition, the claim is proved.
+The above lends significant convenience to our arguments, since it allows us to only construct processes
+matching some claimed member of the fork-convex hull up to finite times, which is typically easy to do in our
+arguments below using just finite fork-convex combinations.
+A.2
+Proofs about the Fork-Convex Hull of Independent Sequential Laws
+We may now proceed with the proofs of the Lemmata omitted from §3.
+Proof of Lemma 4. As detailed in the main text, by taking repeated fork-convex combinations, it follows that
+RT ∈ f-conv(P∞), where
+R1 := P ∞
+1 , RT :=
+�
+RT −1 T −1,0
+−→ P ∞
+T
+�
+,
+where validity of the mixture weight 0 exploits the mutual absolute continuity of laws in P. We conclude by
+Lemma 16.
+Proof of Lemma 5. It suffices to show that for all finite k, P∞
+k
+⊂ f-conv(P∞), since P∗ = �
+k Pk, and Pk ⊂
+Pk+1 for all k.
+For T ∈ N and two laws P, Q on Rd, define the sequential law RP,Q,T as the law of an
+independent sequence {Xt} such that Xt ∼ P for t ≤ T and Xt ∼ Q for t > T , i.e. RP,Q,T =
+�
+P ∞ T,0
+−→ Q∞�
+.
+For T ∈ N, define Pk,T as the set of sequential laws of the form RP,Q,T with P ∈ Pk and Q ∈ P.
+27
+
+We first claim that Pk,T ⊂ f-conv(P ∞). We show this inductively in k. Fix any T , and observe that
+trivially P1,T lies in this fork-convex hull. For k ≥ 2, we may represent each P ∈ Pk as P = αP 1 + (1 − α)P 2
+for some α ∈ [0, 1], P 1 ∈ Pk−1 and P 2 ∈ P. We need to show that for any such P, and any Q ∈ P, RP,Q,T lies
+in the fork-convex hull of P∞. By the induction hypothesis, RP 1,Q,T ∈ f-conv(P∞), and RP 2,Q,T ∈ f-conv(P∞).
+But then define the laws
+S0 := RP 1,Q,T , �Sτ :=
+�
+Sτ−1 τ−1,α
+−→ RP 2,Q,T
+�
+, Sτ :=
+�
+�Sτ
+τ,0
+−→ RP 1,Q,T
+�
+.
+We note that every fork-convex combination above has valid weights since P is m.a.c., and so no density
+process is ever 0. We claim that ST = RP,Q,T .
+Indeed, let p1, p2, q respectively denote the densities (with respect to the standard Gaussian) of P 1, P 2,
+and Q, and let Z1
+t and Z2
+t be the density processes of RP 1,Q,T and RP 2,Q,T respectively. These can be explicitly
+evaluated as
+Zi
+t =
+�
+s≤min(t,T )
+pi(Xs) ·
+t�
+s=min(t,T +1)
+q(Xs),
+where i ∈ {1, 2}, and we note that for u < v, �u
+s=v · = 1. Observe that for each i, and any t1 < T, and t > t1,
+we have
+Zi
+t
+Zi
+t1
+=
+min(t,T )
+�
+s=min(t1,T )+1
+pi(Xs) ·
+t�
+s=min(t,T +1)
+q(Xs).
+We shall inductively claim that for each τ, the density process of Sτ satisfies
+ZSτ
+t
+=
+�
+s≤min(t,τ)
+(αp1(Xs) + (1 − α)p2(Xs)) ·
+min(t,T )
+�
+s=min(t,τ+1)
+p1(Xs) ·
+t�
+s=min(t,T +1)
+q(Xs).
+Indeed, the base claim is trivial since for τ = 0 since S0 = RP 1,Q,T . Assuming the induction hypothesis for
+τ, we observe that since ˜Sτ+1 is a fork-convex combination of Sτ and RP 2,Q,T at time τ, it shares the density
+process of Sτ up to time τ, while after that time the density is a mixture of the two density processes, giving
+Z~Sτ+1
+t
+=
+�
+s≤min(t,τ)
+(αp1(Xs) + (1 − α)p2(Xs))
+×
+
+α
+
+
+
+min(t,T )
+�
+s=min(t,τ+1)
+p1(Xs) ·
+t�
+s=min(t,T +1)
+q(Xs)
+
+
+ + (1 − α)
+
+
+
+min(t,T )
+�
+s=min(t,τ+1)
+p2(Xs) ·
+t�
+s=min(t,T +1)
+q(Xs)
+
+
+
+
+ ,
+where we have used the behaviour of Zi
+t/Zi
+τ above for t ≥ τ + 1.
+Finally, Sτ+1 mixes the above with RP 1,Q,T at time τ + 1 with a mixture weight of 0. This means that the
+suffix law of Sτ+1 beyond the time τ + 2 is exactly equal to the law of RP 1,Q,T , while the prefix up to time
+τ + 1 is left alone. In other words,
+ZSτ
+t
+=
+�
+s≤min(t,τ)
+(αp1(Xs) + (1 − α)p2(Xs)) ·
+τ+1
+�
+s=min(t,τ+1)
+(αp1(Xs) + (1 − α)p2(Xs))
+×
+min(t,T )
+�
+s=min(t,τ+2)
+p1(Xs) ·
+T
+�
+s=min(t,T +1)
+q(Xs).
+The claim follows upon noticing that the first two products can be merged into �
+s≤min(t,τ+1)(αp1(Xs) +
+(1 − α)p2(Xs).
+With this in hand, the argument is straightforward. For any element P ∈ P∞
+k , we note that there exists
+some member of Pk,T , say PT such that the density process of PT matches that of P up to time T . Applying
+Lemma 16 immediately yields the claim.
+28
+
+Proof of Lemma 6. Let P = �{Pt} for any arbitrary sequence of Pt ∈ P.
+We need to show that P ∈
+f-conv(� P). But, since Pt ∈ P for each t, for each t there further exist sequences {P n
+t }n∈N, with each
+P n
+t
+∈ P, such that P n
+t
+→ Pt in L1(Γ).
+Let Q := {P n
+t
+: t, n ∈ N}.
+We note that � Q ⊂ � P
+=⇒
+f-conv(� Q) ⊂ f-conv(� P). Let Q := f-conv(� Q). We shall argue that P ∈ Q.
+Let PT be the sequential law with density process
+ZPT
+t
+=
+��
+s≤t ps(Xs)
+if t ≤ T
+�
+s≤T ps(Xs) · �
+T <s≤t p1
+s(Xs)
+if t > T .
+If we can show that for each T , PT ∈ Q, then the claim will follow, since PT → �{Pt} as in the argument of
+Lemma 16, and since Q is closed under the relevant notion of convergence.
+We shall show this inductively. Let P1,n be a sequential law with density Z1,n
+t
+:= pn
+1(X1)·�
+s>min(1,t) p1
+s(Xs).
+Notice that P1,n ∈ � Q ⊂ Q for every n. Further,
+∆(P1,n, P1) ≤ ∥Z1,n
+1
+− ZP1
+1 ∥L1(Γ) → 0.
+Thus P1 ∈ Q.
+Now suppose that PT −1 ∈ Q for some T ≥ 2. For T, n ∈ N, define Qn as the sequential law of density ratio
+ZQn
+t
+:=
+
+
+
+
+
+�
+s<t p1
+s(Xs)
+t < T
+ZQn
+T −1 · pn
+T (XT )
+t = T
+ZQn
+T · �t
+s=T +1 p1
+s(Xs)
+t > T
+.
+It trivially follows that Qn ∈ � Q ⊂ Q for all n. Now, define
+PT,n =
+�
+PT −1 T −1,0
+−→ Qn�
+,
+which is valid since each P n
+t and Pt has are mutually absolutely continuous. But ZPT,n
+t
+= ZPT −1
+t
+= ZPT
+t
+for
+t ≤ T − 1, and for t ≥ T,
+ZPT,n
+t
+− ZPT
+t
+= ZPT
+T −1 · (pn
+T (XT ) − pT (Xt)) ·
+t�
+s=T +1
+ps(Xs).
+It follows that
+∥ZPT,n
+t
+− ZPT
+t ∥ =
+�
+0
+t < T
+∥P n
+T − PT ∥L1(Γ)
+t ≥ T ,
+and therefore, ∆(PT,n, PT) ≤ ∥P n
+T − PT ∥L1(Γ) → 0. By closeness of Q, we conclude that PT ∈ Q.
+A.3
+Proof of Lemma 8
+Proof of Lemma 8. Fix an m ∈ N. Since E has positive mass and is measurable, there exists an open set
+O ∈ (Rd)t such that O ⊃ E and Lebdt(O) ≤ (1 + 1/m)Lebdt(E). Observe here that ‘most’ of the mass of O
+lies within E.
+Since O is open, there exists a sequence of disjoint open rectangles Ri in (Rd)t such that � Ri ⊂ O ⊂ � Ri,
+and
+Lebdt
+��
+Ri
+�
+=
+�
+Lebdt(Ri) = Lebdt(O).
+Further, since most of the mass of O lies in E, we conclude that there exists at least one i such that
+Lebdt(Ri) > 0
+and
+Lebdt(E ∩ Ri) ≥
+m
+m + 1Lebdt(Ri).
+Indeed, otherwise we would have
+Lebdt(E) = Lebdt(E ∩ O) =
+�
+Lebdt(E ∩ Ri) <
+m
+m + 1
+�
+Lebdt(Ri) ≤
+m
+m + 1 · m + 1
+m
+Lebdt(E),
+which is impossible.
+29
+
+A.4
+Technical Aspects of Fork-Convex Hulls and Our Triviality Argument
+We comment on some technical aspects of the argument underlying the non-existence of nontrivial NSMs.
+Specifically, we discuss the necessity of our definition of nontriviality, and the m.a.c. condition repeatedly used
+in the argument, how the argument can be extended to consider log-concave laws over bounded sets, and
+finally issues that arise when one tries to relax the definition of fork-convex combinations to handle support
+mismatch.
+Going beyond almost sure triviality.
+The main text defines trivial NSMs (and NMs) as those that are
+Γ-almost surely non-increasing (respectively, constant). Could one instead show that there are no nontrivial
+L∞-NSMs in the stronger sense that such processes must be non-increasing (as opposed to only almost surely
+non-increasing)? This turns out to be impossible, as witnessed by the following process
+Mt :=
+1
+1 − 1{∃(t1, t2, t3, t4) ∈ [1 : t]4 : Xt1 = Xt2, Xt3 = Xt4, Xt1 ̸= Xt3}.
+Since log-concave measures can have at most one atom (due to unimodality), it follows that {Mt} is an
+L∞-martingale (indeed, it is almost surely just a constant 1, as stated by the theorem). However, Mt does
+diverge to ∞, and this occurs almost surely against any i.i.d. sequential law which has at least two atoms,
+for instance, a coin flip process. This means that while it may not be possible to reject processes with a
+Lebesgue density using test martingales, it is possible to reject atomic processes. Structurally, this example
+has to do with the fact that one cannot approach point masses in an L1 sense using measures with density.
+Therefore, although L∞-NSMs must also be NSMs for independent processes with densities, this does not
+extend to sequences drawn from distributions with atoms. In another sense, this issue is the same as the
+problem discussed below regarding loss of the NSM property under extensions of fork-convex combinations of
+laws with support mismatch, in that two laws with distinct single atoms each have parts that are mutually
+singular.
+The role of the mutual absolute continuity condition on P.
+The definition of fork-convex combinations
+of two laws P and Q at time s involves the ratio of density processes ZP
+s/ZQ
+s. This ratio must indeed appear, as
+can be seen from the algorithmic viewpoint of §3 to account for the fact that if R is the fork-convex combination,
+then the prefix law R|s = P|s. However, if ZQ
+s = 0, i.e. if for {Xt} ∼ R, the prefix Xs
+1 lies in a set that is almost
+surely impossible under Q, then the above ratio is meaningless. This observation underlies the condition that
+if ZQ
+s = 0, then the mixture weight h must be exactly 1.
+Our argument ultimately asserts that any law of the form �{Pt} lies in f-conv(P∞).
+However, our
+constructions to demonstrate this fact rely on setting h = 0 in order to generate switches between different
+laws in P. Our assumption of mutual absolute continuity is to enable precisely this flexibility without running
+into the issue discussed in the previous paragraph.
+The role of Gaussians in our argument.
+Since we used the density of the Gaussians in order to show
+that L∞-NSMs must also be � D-NSMs, it behooves us to ask how essential L ⊃ G is to the main point of the
+result.4 In the argument, Gaussians play two roles: firstly, since all Gaussians are supported on the entirety
+of the domain, this class is m.a.c., and we can flexibly take fork-convex combinations. Secondly, the triviality
+of Gaussian NSMs follows since mixtures of Gaussians are L1-dense in the set of densities on the reals. Any
+subset of L that satisfies these two properties will suffice for our purposes.
+Extending the argument to log-concave laws on subsets of Rd.
+We finally observe that our argument
+extends in a straightforward manner to log-concave laws on restricted subsets of the reals: for a bounded
+convex set K, define LK to be log-concave densities supported on K.
+Then all L∞
+K -NSMs are trivial, in
+the sense that they are almost surely nonincreasing with respect to the reference measure (Unif(K))∞. This
+follows because truncated Gaussians are again dense and supported on the entirety of the domain K.
+4Notwithstanding that the result is interesting in its own right for Gaussians, which tells us that there is a simple, and very
+natural, parametric family that cannot be tested via nonnegative supermartingales.
+30
+
+To see this, first observe that if γ := � αiφi is a mixture of Gaussians, then for any K of nonzero Lebesgue
+mass, the truncation γ|K is also a mixture of truncated Gaussians. Indeed, define θi =
+�
+K φi. Then
+γ|K(x) =
+�
+αi
+� αiθi
+φi(x) · 1{x ∈ K} =
+�
+αiθi
+� αiθi
+φi|K(x).
+Now, let p be any density supported on K, and let γn → p be a sequence of mixtures of Gaussians converging
+so that dn :=
+�
+|p − γn| → 0. Then, defining πn =
+�
+Kc γn, we have
+�
+|p − γn|K| =
+�
+K
+|p(1 − πn) − γn|
+1 − πn
+≤
+�
+K
+|p − γn|
+1 − πn
++
+�
+K
+πnp
+1 − πn
+≤ πn +
+�
+|p − γn|
+1 − πn
+.
+Further, since p is supported on K, πn =
+�
+Kc γn ≤
+�
+Kc γn +
+�
+K |p − γn| = dn. Therefore,
+TV(p, γn|K) ≤
+2dn
+1 − dn
+→ 0.
+But this means that we can run the entire argument of §3 but with Gaussians truncated over K, and draw
+the same conclusion.
+Can we extend nontrivial fork-convex combinations to all laws?
+As we discussed above, due to the
+“ZQ
+T = 0 =⇒ h = 1” condition in the definition of fork-convex combinations, it is not possible to take arbitrary
+fork-convex combinations between sequential laws. In the extreme case of P = P ∞ and Q = Q∞ for P, Q that
+have disjoint support, the only possible fork-convex combinations are mixtures of the form αP ∞ + (1 − α)Q∞.
+While this technicality did not pose a serious issue for the current paper, this situation is quite unsatisfying
+in general. After all, the algorithmic view of fork-convex combinations is very natural, and extends to such
+disjoint support situations easily.
+One can formalise this algorithmic picture by exploiting conditional densities. For a sequence of (appro-
+priately measurable) maps kP
+t : (Rd)t−1 × Rd → R≥0, denoted kP
+t (xt|xt−1
+1
+), we say that {kP
+t } is the conditional
+density process of P if for each xt−1
+1
+, kt(·|xt−1
+1
+) is a density with respect to Γ, and for any t, A ∈ Ft,
+P(Xt
+1 ∈ A) =
+�
+A
+�
+s≤t
+ks(xs|xs−1
+1
+)Γ(dxt
+1).
+More generally, we can define a similar notion via Markov kernels. We observe that, by definition, it holds
+that if P has a conditional density process, then for any t and Γ-almost every xt
+1 that
+ZP
+t (xt
+1) =
+�
+s≤t
+ks(xs|xs−1
+1
+).
+Using the above characterisation, we can give the following natural extended definition of fork-convex
+combinations: for two sequential laws P, Q with conditional density processes {kP
+t}, {kQ
+t} respectively, a law R
+is a fork-convex combination of P and Q at time T with FT -measurable weight h if
+ZR
+t =
+��
+s≤t kP
+s(xs|xs−1
+1
+)
+t ≤ T
+�
+s≤T kP
+s(xs|xs−1
+1
+) ·
+�
+h �t
+s=T +1 kP
+s(xs|xs−1
+1
+) + (1 − h) �t
+s=T +1 kQ
+s(xs|xs−1
+1
+)
+�
+t > T ,
+(4)
+the difference being that we now do not impose the restriction that h = 1 if ZQ
+T = 0. Simplistically, this is
+possible since we are never dividing by the potentially null ZQ
+T , and more formally, this is considering the
+formal ratio ZQ
+t /ZQ
+T , which is interpreted in the natural way as � kQ
+s(xs|xs−1
+1
+). The above extended definition
+genealizes our previous definition of fork-convex combinations, and we can extend the same to the fork-convex
+hull and its closure.
+While the density process above is a perfectly sound mathematical object, such an extension is not fruitful
+because of a failure to preserve the NSM property under these extended fork-convex combinations in general.
+To illustrate why the above extended definition fails to maintain the NSM property (unlike the restricted
+one used in the paper), consider the following example.
+31
+
+Example 1. P = (Unif(0, 1))∞ and Q = (Unif(1, 2))∞, and the process
+Mt :=
+�
+2
+∃s1, s2 ≤ t : Xs1 ∈ (0, 1), Xs2 ∈ (1, 2)
+1
+otherwise
+.
+This process is an NSM (indeed, a martingale) under both P, Q. However, under any nontrivial fork-convex
+combination of these two laws, this process must start at 1, and with positive probability grow to 2 but never
+fall, and thus cannot be a supermartingale.
+Under the hood, the issue in the example above arises due to the fact that under the extended definition,
+for t ≥ T +1, {ZR
+t > 0} = {ZP
+t > 0}∪{ZP
+T > 0, �t
+T +1 ks(Xs|Xs−1
+1
+) > 0}, but the NSM property of {Mt} under
+P or Q only controls the conditional expectations of MtZP
+t 1{ZP
+t > 0} and MtZQ
+t 1{ZQ
+t > 0} under Γ, which
+leaves the conditional behaviour of MtZR
+t uncontrolled when R places mass on events that are null under one
+of these laws.
+It should be noted that in the above example there is a version of the process {Mt}, i.e., a process {�
+Mt}
+such that P(∀t, Mt = �
+Mt) = Q(∀t, Mt = �
+Mt) = 1, but such that {�
+Mt} is a martingale even under extended
+fork-convex combinations. Concretely this process is just �
+Mt = 1. One may thus wonder if this phenomenon
+holds true in greater in generality: is it the case that if {Mt} is an NSM under P and Q, then there is a version
+{�
+Mt} of it (under P and Q) such that {�
+Mt} is an NSM against any extended fork-convex combination of P
+and Q, without the restriction “ZQ
+T = 0
+=⇒
+h = 1”? This turns out also to be impossible in general, as
+demonstrated by the following example.
+Example 2. Let P = Unif(0, 1)∞ and Q = Unif(0, 1/2)∞. Define ρt = 1{Xt ∈ (0, 1/2)} for t ≥ 1, and ρ0 = 0.
+Let {Nt} be an adapted process such that
+Nt =
+
+
+
+
+
+1
+ρt−1 = 1
+3/2
+ρt−1 = 0, ρt = 1
+1/2
+ρt−1 = 0, ρt = 0
+.
+Finally define Mt = �
+s≤t Nt. It is easy to check that Mt is an NM under both P and Q.
+Now suppose R is an extended fork-convex combination of P, Q at time T ≥ 1 with mixture weight h < 1.
+This means that with probability 1 − h, it holds that Xt ∈ (0, 1/2) with certainty for all t ≥ T + 1. As as result,
+we can explicitly compute that
+E[NT +1|FT ] = ρT + (1 − ρT ) ((1 − h) · 3/2 + h(1/2 · 3/2 + 1/2 · 3/2)) =
+�
+1
+ρT = 1
+1 + (1 − h)/2
+ρT = 0 ,
+and so as long as h < 1, E[NT +1|FT ] > 1 if ρT = 0, and therefore {Mt} violates the NSM property under R
+at the time T + 1. Note here that it is hard to construct any nontrivially different version of the above process
+since the law of P dominates that of Q.
+In light of the above discussion, generalised definitions of fork-convex combinations are at loggerheads with
+maintaining the NSM property these combinations. Of course, since our purpose in using fork-convexity is to
+assert the triviality of NSMs over large classes of sequential laws, this latter property is essential to maintain for
+such statistical applications. At the same time, while the restricted original definition does maintain the NSM
+property, the included restriction is unsatisfying, and in conflict with the algorithmic intuition underlying
+the idea of these combinations. Finding an appropriate generalised definition of fork-convex combinations
+that abstains from imposing these support conditions, but nevertheless retains NSM closure under the NSM
+property is an interesting, and challenging, question left for future work.
+B
+Proofs of Consistency and Power Analysis
+Recall the notation σt(P) := �
+s≤t log p(Xs) − log ˆpt(Xs). The main arguments of this section control the
+behavious of σt(P), in particular arguing that if the Hellinger distance of P from log-concavity is large, then
+32
+
+σt(P) must eventually grow linearly. We show this in asymptotic and nonasymptotic regimes in §B.1 and §B.2
+respectively.
+Corollary 11 and Corollary 15 each relies on further control on the behaviour of ρt(E ; p) = �
+s≤t log p(Xs)−
+log ˆqs−1(Xs) when p is a bounded Lipschitz law on the unit box. This argument is left to §B.3.
+B.1
+Proof of Consistency
+Our arguments rely on the following bracketing tail estimate, developed by Wong and Shen to analyse the
+behaviour of sieve-based maximum likelihood estimates [WS95, Thm. 1]. The estimate involves the bracketing
+entropy of a class of laws Q under the Hellinger metric. We refer the reader to the text of Van der Vaart and
+Wellner [VW96] for a thorough introduction, and give a brief account.
+A bracket [u, v] is defined by two functions u(x), v(x) such that u(x) ≤ v(x) for all x, and consists of the
+set of all functions f such that u(x) ≤ f(x) ≤ v(x) for all x. Since we shall only be interested in functions
+that are densities, we may restrict attention to nonnegative functions. The Hellinger size of such a bracket
+[u, v] is defined as |[u, v]| = ∥√u − √v∥2/2. We say that a class of distributions Q is bracketed by {[ui, vi]}N
+i=1
+if for all Q ∈ Q, there exists an i such that q ∈ [ui, vi], where recall that for a distribution Q, we denote its
+density by q. Note that this bracketing is typically “improper”, i.e., ui, vi will generally not lie in Q (because
+q integrates to one, and so its lower bracket u will integrate to less than one, and its upper bracket v will
+integrate to more than one). The Hellinger bracketing entropy of Q at scale ζ is the logarithm of the most
+parsimonious bracketing of Q by brackets of size at most ζ, i.e.
+H[](Q, ζ) := inf{log N : Q has an N-sized Hellinger bracketing at scale ζ}
+Note, of course, that bracketing entropies are nonincreasing in ζ.
+Lemma 17. (Simplification of [WS95, Thm. 1]) For a class of distributions Q and a natural number t, define
+εt as the smaller number ε such that
+� √
+2ε
+ε2/28
+�
+H[](Q, ζ/10)dζ ≤ 2−11√
+tε2.
+For every t and ε ≥ εt, it holds that for any law P such that dH(P, Q) ≥ ε, we have
+P ⊗t
+
+ inf
+q∈Q
+�
+s≤t
+log p(Xs) − log q(Xs) ≤ tε2/24
+
+ ≤ 4 exp
+�
+−Ctε2�
+,
+where C > 2−14 is a constant.
+Informally, if P is far enough from Q in the Hellinger metric (where far enough is determined by the
+bracketing entropy of Q), then it is exponentially unlikely (in the sample size) for the maximum log-likelihood
+under Q to be linearly close to the log-likelihood under P. Exploiting this observation in our context requires
+us to argue that eventually, the log-concave MLE ˆpt must lie in a set with small entropy. To this end, we
+appeal to the following result due to Dunn et al., which extends the convergence analysis of Cule & Samworth
+[CS10].
+Lemma 18. [DGWR21, Lem. 1] Consider any distribution P ∈ D1, not necessarily log-concave. For any
+η > 0, there exists a bracket [uη, vη] of size at most η that contains the log-concave projection LP , and
+eventually also contains the log-concave MLE ˆpt: P ∞(∃t0 : ∀t ≥ t0, ˆpt ∈ [uη, vη]) = 1.
+In words, the lemma states that for large enough t, the log-concave MLE ˆpt is certain to lie in a very
+small bracket around the log-concave projection LP of the true distribution P. With this in hand, we are in
+a position to show Lemma 12, the main statement underlying the proof of Theorem 10.
+Proof of Lemma 12. Let ε := dH(P, LP ) > 0. Define ηε = ε2/211. Using Lemma 18, we know that there exists
+a bracket [u∗, v∗] such that |[u∗, v∗]| ≤ ηε and, almost surely, ˆpt ∈ [u∗, v∗] for all large enough t. But observe
+33
+
+that H[]([u∗, v∗], ε2/211) = 0, since the size of [u∗, v∗] is already ηε. Further, since LP ∈ [u∗, v∗], by the triangle
+inequality,
+dH(P, [u∗, v∗]) =
+inf
+Q∈[uη,vη] dH(P, Q) ≥ dH(P, LP ) − |[u∗, v∗]| ≥ ε(1 − 2−11) ≥ ε ·
+�
+24/25.
+Let us define
+�σt(P) :=
+inf
+q∈[u∗,v∗]
+�
+s≤t
+log p(Xs) − log q(Xs).
+By exploiting the above observations, Lemma 17 yields that for every t,
+P ∞ �
+�σt(P) ≤ tε2/25
+�
+≤ 4 exp
+�
+−Ctε2�
+.
+Note further that if ˆpt ∈ [u∗, v∗], then since ˆpt is a maximum likelhood estimate, it must hold that
+σt(P) = �σt(P). Let Es := {∀t ≥ s, ˆpt ∈ [u∗, v∗]} be the event that ˆpt lies in the small bracket after time s,
+and At := {σt(P)/tε2 ≥ 1/25} be the event that σt(P) is larger than tε2/25.
+By Lemma 17, for every fixed time s and t ≥ s,
+P ∞(Ac
+t ∩ Es) ≤ 4 exp
+�
+−Ctε2�
+,
+and since this upper bound is summable, by the Borel-Cantelli Lemma
+0 = P ∞
+�
+lim sup
+t
+(Ac
+t ∩ Es)
+�
+= P ∞
+�
+(lim sup
+t
+Ac
+t) ∩ Es
+�
+,
+and so for any time s,
+P ∞(lim sup
+t
+Ac
+t) ≤ P ∞(lim sup
+t
+Ac
+t ∩ Es) + P ∞(Ec
+s) = P ∞(Ec
+s).
+By Lemma 18, ˆpt must eventually almost surely fall in [u∗, v∗], lims→∞ P ∞(Ec
+s) → 0. Further notice that
+lim sup
+t
+Ac
+t = {σt(P)/tε2 < 1/25 infinitely often} = {lim inf σt(P)/tε2 < 1/25}.
+Putting the observations together, we conclude upon sending s → ∞ that
+P ∞(lim inf σt(P)/tε2 < 1/25) ≤ lim
+s→∞ P ∞(Ec
+s) = 0.
+B.2
+Proofs Underlying the Power Analysis
+We shall begin by stating the key lemmata underlying our argument, which exploit our bracketing entropy
+control from Lemma 13 along with results in the literature that bound the maximum value attained by a
+log-concave density in order to make the same effective. We then prove the main result, and conclude by
+proving Lemma 13.
+B.2.1
+Controlling the Maximum Value Attained by the Log-Concave MLE
+The rate analysis quantitatively exploits Lemma 17. To do so, we first need bracketing entropy bounds for
+log-concave laws, which is precisely the subject of Lemma 13. We recall that this controls the bracketing
+entropy of the class Ld,B of log-concave laws with densities supported on [−1, 1]d that are bounded from above
+by B, showing that
+H[](Ld,B, ζ) = �O((B/ζ)max(d/2,(d−1))).
+The role of B in the above is quantitatively unimportant as long as this constant does not scale with
+relevant parameters. This fact is assured for log-concave laws with near identity covariance. Intuitively, since
+the covariance is lower bounded in all directions, the laws cannot concentrate too much, and thus the value of
+the density at the mode cannot be too large. This observation is encapsulated in the following result, which
+follows trivially from the work of Kim & Samworth.
+34
+
+Lemma 19. [KS16, Cor. 6] Let Lγ
+d denote the set of log-concave laws distributed on [−1, 1]d with covariances
+lower bounded in the positive semidefinite order by γI. Then there exists a dimension dependent constant Cd
+such that for any f ∈ Lγ
+d,
+max
+x∈[−1,1]d f(x) ≤ γ−d/2Cd.
+Of course, our bounds in Theorem 14 depend on ∆P , which roughly speaking only controls that the
+covariance of the underlying law P. The relevance of this quantity arises from the following observation, due
+to Barber and Samworth.
+Lemma 20. [BS21, Cor. 8] Let P ∈ D1 be a law supported on [−1, 1]d such that
+∆P :=
+min
+v:∥v∥=1 Ep[|⟨v, X − Ep[X]⟩|] > 0.
+Then there exists a dimension dependent constant cd such that Cov(LP ) ⪰ cd∆2
+P I. Further, there exists a
+dimension-independent constant C such that for any t ≥ 2Cd3/∆2
+P , it holds with probability at least 1 −
+2 exp
+�
+−Ct∆2
+P /d2�
+that Cov(ˆpt) ⪰ cd∆2
+P
+4
+I for the log-concave MLE ˆpt.
+Proof of Lemma 20. The first observation is a direct restatement of Lemma 7 of Barber and Samworth. The
+second statement follows from the fact that over v : ∥v∥ = 1, v �→ ⟨v, X − EP [X]⟩ is bounded by 2
+√
+d, and
+is clearly continuous in v. Thus exploiting standard subGaussian concentration results over the unit ball, it
+follows that with probability at least 1−2 exp
+�
+−Ct∆2
+P /d2�
+, it holds that for the empirical law pt = 1
+t
+�
+s≤t δXs,
+min
+v:∥v∥=1 Ept[|⟨v, X − Ept[X]⟩|] ≥ ∆P /2.
+But notice that ˆpt = Lpt, from which the claim follows by the first part.
+Merging Lemmas 19 and 20 immediately yields the following observation, which serves as a concrete bound
+for the scale of B we need to employ in Lemma 13.
+Lemma 21. There exists a constant Cd depending only on d such that for any t ≥ 2Cdd3/∆2
+P , it holds with
+probability at least 1 − 2 exp
+�
+−Cdt∆2
+P /d2�
+that
+max
+x∈[−1,1]d ˆpt(x) ≤ Cd∆−d
+P .
+Proof. Employing Lemma 19, we observe that {max ˆpt ≤ (cd∆2
+P /4)−d/2} ⊂ {Cov(ˆpt) ⪯ cd∆2
+P /4I}, and the
+latter has probability at least 1 − 2 exp
+�
+−t(C∆2
+P /d2)
+�
+for t ≥ 2Cd3/∆2
+P . Take Cd = max(C, c−d/2
+d
+).
+B.2.2
+Proof of Bounds on Rejection Times
+With the above in hand, we may proceed with the main argument.
+Proof of Theorem 14. Recall the definition σt := �
+s≤t log p(Xs) − log ˆpt(Xs). We shall first lower bound σt
+with high probability.
+Let B be a quantity that we will choose later. Let εt denote the solution to the fixed point equation from
+Lemma 17, instantiated with the bracketing entropy of Ld,B. Further, let define the event
+Et := {ˆpt ∈ Ld,B}.
+For any t, provided that such that εt ≤ dH(p, L) and ˆpt ∈ Ld,B, Lemma 17 yields that
+σt ≥
+inf
+q∈Ld,B:dH(p,q)≥dH(p,L) log
+�
+s≤t
+p(Xs)
+q(Xs) ≥ td2
+H(p, L)
+24
+(5)
+with probability at least 1 − exp
+�
+−Ctd2
+H(p, L)
+�
+− P ∞(Ec
+t).
+35
+
+In the rest of the proof, we will determine the range of t that leads to a small enough value for εt to
+ensure that the condition εt ≤ dH(p, L) is met and, at the same time, control P ∞(Ec
+t). To this end, we deploy
+Lemma 13. First, observe that for d ≥ 3 and for any positive constants c and C
+� Cε
+cε2
+�
+˜O(Bd−1ζ−(d−1))dζ = �O(B(d−1)/2ε−(d−3)).
+Note that polylogarithmic terms do not affect the main growth of the integral.5 Therefore, solving the fixed
+point equation
+�O(B(d−1)/2ε−(d−3)) = ε2t1/2,
+we obtain that for d ≥ 3
+εt(B) = �O(B1/2t−1/2(d−1)),
+where we highlight the dependence on the as yet undetermined quantity B.
+A similar argument using the entropy bound ζ−d/2 yields εt(B) = �O(Bd/(d+4)t−2/(d+4)) for d ∈ {1, 2}.
+Now define
+T1(B) = inf{t : εt(B) ≤ dH(p, L)}
+and observe that
+T1(B) =
+� �O(B(d−1)(dH(p, L))−2(d−1))
+d ≥ 3
+�O(Bd/2(dH(p, L)−(4+d)/2))
+d ∈ {1, 2} .
+Finally, by Lemma 21, for B ≥ Cd∆−d
+P
+and t ≥ T2 := C∆2
+P /d2 the probability of the event Et is at least
+1 − 2 exp
+�
+−tC∆2
+P /d2�
+. Let us set B∗ = Cd∆−d
+P
+and let T0 := max(T1(B∗), T2). We obtain that the lower
+bound
+σt ≥ td2
+H(p, L)
+24
+holds with probability at least 1−C exp
+�
+−tcd2
+H(p, L)
+�
+−C exp
+�
+−tc∆2
+P /d2�
+for t ≥ T0. Now, observe that at any
+time t ≥ max(T0, 600 log(1/α)
+d2
+H(p,L)
+), it holds with probability at least 1−πt−C exp
+�
+−td2
+H(p, L)
+�
+−C exp
+�
+−Ct∆2
+P /d2�
+that
+log Rt = σt − ρt ≥ td2
+H(p, L)
+600
+≥ log(1/α),
+and thus the probability that the rejection time τα := inf{t : Rt ≥ 1/α} exceeds the above bound is bounded
+by πt + C exp
+�
+−td2
+H(p, L)
+�
++ C exp
+�
+−Ct∆2
+P /d2�
+.
+B.2.3
+Proof of Bracketing Entropy Bound on Log-Concave Laws
+We proceed to show Lemma 13. We note that the upper bound for d ≤ 3 was shown by Kim and Samworth
+[KS16]. Below we focus on d ≥ 4. We shall exploit two existing results in the literature regarding convex sets
+and functions. The first is essentially due to Bronshtein (also see [KDR19, Lem. 3]).
+Lemma 22. [Bro76] Let Kd denote the collection of convex sets in [−1, 1]d. For any ζ > 0, there exists a
+collection of pairs of convex sets Kd,ζ ⊂ Kd × Kd with log |Kd,ζ| = O(ζ−(d−1)/2) such that
+• Every (K, K) ∈ Kd,ζ satisfies Lebd(K \ K) ≤ ζ.
+• For every K ∈ Kd, exists (K, K) ∈ Kd,ζ satisfying K ⊂ K ⊂ K.
+In other words, the bracketing entropy of convex sets under the set difference metric is controlled at rate
+(d − 1)/2. Importantly, the bracketing demonstrated above is proper. This result may be extended to the
+following bracketing entropy bound on convex functions as by Gao and Wellner.
+Lemma 23. [GW17, Thm. 1.5] Let K be a convex set in [−1, 1]d, and let CK,B be the set of convex functions
+upper bounded by B over K. Then the L2(K) bracketing entropy of CK,B at scale ζ is bounded as O((B/ζ)(d−1)).
+5This can be seen by iterating the relation
+�
+xn logm x = xn+1 logm x
+n+1
+−
+m
+n+1
+�
+xn logm−1(x).
+36
+
+Above, the L2(K) metric is the usual L2 distance ∥f − g∥L2(K) = (
+�
+K(f − g)2dx)1/2, and the L2(K)
+bracketing entropy is the bracketing entropy when the size of a bracket [u, v] is |[u, v]| = ∥u − v∥L2(K).
+With the above in hand, we may proceed with the proof.
+Proof of Lemma 13. For any log-concave law f, let S := {x ∈ Rd : f(x) ≥ ζ3} = {x ∈ Rd : log f(x) ≥ 3 log ζ}.
+Since f is log-concave, the set S is convex. As a result, by Lemma 22, there exists some convex set ˜S ∈ Kd,ζ2/B
+such that Leb(S \ ˜S) ≤ ζ2/B and ˜S ⊂ S. Let ˜C ˜S,ζ,B denote a ζ-bracketing of convex functions bounded by
+B on ˜S. Since, on ˜S, the function − log f is convex and is upper bounded by − log B, by Lemma 23 there
+exists a bracket [−u, −l] ∈ ˜C ˜S,ζ/B,log B/ζ3 such that, on ˜S, l ≤ log f ≤ u, and
+�
+˜S(u(x) − l(x))2dx ≤ ζ2/B2.
+Note that, on ˜S, f is lower bounded by −3 log ζ and that, without loss of generality, we may assume that
+supx∈ ˜S u(x) ≤ log B, since this is already a pointwise upper bound on f.
+Next, we construct the functions
+x ∈ [−1, 1]d �→ U(x) :=
+
+
+
+
+
+eu(x)
+x ∈ ˜S
+B
+x ∈ S \ ˜S
+ζ3
+x ∈ [−1, 1]d \ S
+,
+x ∈ [−1, 1]d �→ L(x) :=
+
+
+
+
+
+el(x)
+x ∈ ˜S
+ζ3
+x ∈ S \ ˜S
+0
+x ∈ [−1, 1]d \ S
+.
+Observe that U ≥ f ≥ L on [−1, 1]d. Furthermore, for ζ < 2−d,
+�
+(
+√
+U −
+√
+L)2dx =
+�
+˜S
+(eu(x)/2 − el(x)/2)2dx +
+�
+S\ ˜S
+Bdx +
+�
+[−1,1]d\S
+ζ3dx
+≤
+�
+˜S
+eu(x)(1 − eu(x)−l(x)/2)2dx + B · ζ2/B + 2dζ3
+≤
+�
+˜S
+B2(u(x) − l(x))2/4 dx + 2ζ2
+≤ B2 · ζ2/B2 + 2ζ2 = 3ζ2,
+where we have exploited the fact that z �→ ez/2 is Lipschitz on [−∞, log C], with derivative bounded by
+e(log B)/2/2 =
+√
+B/2 to argue that e(u(x)−l(x))/2 − e0 ≤
+√
+B|u(x) − l(x) − 0|/2.
+Since this construction can be carried out for any f, we conclude that we can construct a bracketing cover
+of Ld,B at scale O(ζ) as the union of the bracketing covers of convex functions on each of the smaller sets in
+Kd,ζ2/B. By Lemmas 22 and 23, the size of this cover is
+exp
+�
+O((B/ζ)d−1)
+�
+· exp
+�
+O((log B/ζ3)(B)/ζ)d−1�
+= exp
+�
+�O((B/ζ)d−1)
+�
+,
+and the claim now follows. Let us again observe that the resulting cover is improper, in that the maps U(x)
+and L(x) are not log-concave.
+B.3
+Regret Control for Bounded Lipschitz Laws on the Unit Box
+As this subsection demonstrates, both Corollaries 15 and 11 rely on arguing that laws in DBox,Lip,B can be
+estimated in a low-regret manner online. We argue this by exploiting the following result, which follows as a
+simplification of the results of Wong and Shen on sieve estimators.
+Lemma 24. (Adaptation of [WS95, Cor. 1 & Thm. 6])For every P ∈ DBox,Lip,B and t ≥ 1, there exists a
+sieve MLE ˆq(·) = ˆq(·; Xt
+1) and a constant A > 1 depending only on B such that for every ζ ≥ ζt,
+P ∞
+�
+KL(p∥ˆq) > 1
+Aζ2 log(1/ζ)
+�
+≤ A exp
+�
+−t
+ζ2
+A log(1/ζ)
+�
+,
+where ζt = �O(t−1/2(d+2)).
+37
+
+Proof. The cited results of Wong and Shen apply because densities of laws in DBox,Lip,B are uniformly upper
+bounded. This directly yields the entirety of the statement, barring the scale bound on ζt. This scale is
+determined by the same entropy integral fixed point equation that appears in Lemma 17, and for this instance,
+the bound can be derived by using the standard fact that the Hellinger bracketing entropy of Lipschitz functions
+on a box at scale η are controlled as O(η−(d+1)) [Vaa94].
+The sieve estimators in this result can be taken with a fair bit of lassitude. In particular, one explicit choice
+is to construct for each ζ > 0 a bracket of the class DBox,Lip,B at scale ζ, and choose a representative density
+within each bracket of the class. The sieve MLE then involves choosing a ζ at each time, and estimating the
+law as the maximiser of likelihood amongst the aforementioned representative densities. Importantly for us,
+the lower brackets in these bracketings can be taken to be uniformly larger than 1/B, and the upper brackets
+smaller than B, since p ∈ [1/B, B], and as a result the sieve estimates are uniformly bounded between 1/B
+and B.
+Below we first show Corollary 15 using the above results, and then show Corollary 11 follows as a simple
+consequence of this argument.
+Proof of Corollary 15. As argued in the main text, the expected rejection time is bounded as E[τ] ≤ � πt +
+O(T0), where T0 = o(dH(p, L)−2(d+3)). We thus only need to show that a sequence πt exists such that � πt is
+appropriately small, and that for any t,
+P ∞
+� ρt(E ; p)
+td2
+H(p, L) ≥ 1
+25
+�
+≤ πt,
+where p ∈ DBox,Lip,B and E are sieve estimators. We proceed to do so below.
+For succinctness, we shall define ε = dH(p, L). Let A be the constant from Lemma 24, and set
+T1 := min{t : ζ2
+t log(1/ζt)/A < ε2/200, ζt < 1/√e}.
+Further let
+ζ(ε) := max{ζ ∈ [0, 1/√e] : ζ2 log(1/ζ) ≤ Aε2/200}.
+In the subsequent proof, we shall use Lemma 24 with ζ = ζ(ε) ≥ ζT1. To this end, we note that if ζ(ε) <
+1/√e ⇐⇒ Aε2/200 < 1/2e, and in this case the equality ζ(ε)2 log(1/ζ(ε)) = Aε2/200 holds. From this, we
+may derive6 that ζ(ε)2 > aε2/ log(1/ε) for some small enough constant a, and so that the exponent of the
+upper bound of Lemma 24 is
+ζ(ε)2
+A log(1/ζ(ε)) = 400ζ4
+A2ε2 ≥
+ε2
+A′ log(1/ε)
+for some large enough constant A′. We shall also assume that A′ ≥ max(1, A).
+Let E be a choice of sieve estimators such that for every t, x ∈ [−1, 1]d, ˆqt−1(x) ∈ [1/B, B], which can be
+ensured due to the discussion above. Notice, by the independence of the data {Xt}, that for any t,
+E[log p(Xt)/ˆqt−1(Xt)|Ft−1] = KL(p∥ˆqt−1).
+Let θ ∈ (0, 1) and M ≥ 0 be two parameters of argument that we shall set later. Let us consider the case
+of t = T1 + τ for some τ ≥ MT1.
+Since for each τ > 0, ζT1+θτ ≤ ζT1 ≤ ζ(ε), the bound of Lemma 24 is effective at each time s ∈ [T1 + θτ :
+T1 + τ] with ζ = ζ(ε). As a result, applying Lemma 24 to each s in this range, and exploiting the behaviour
+of ζ(ε)2 established above,
+P ∞(KL(p∥ˆqs−1) > ε2/200) ≤ A′ exp
+�
+−s
+ε2
+A′ log(1/ε)
+�
+.
+6This equation is equivalent to x log x = y for x = ζ(ε)2, y = Aε2/100 in the range 0 < x < 1/e.
+The claim follows
+by noting that the map x �→ x log(1/x) is monotonically increasing on [0,1/e], and verifying that for y ∈ [0, 1/2e],
+y
+2 log(1/y) ·
+log(2 log(1/y)/y) < y. Indeed, this inequality is equivalent to arguing that log(2 log(1/y)) < log(1/y)
+⇐⇒
+y log(1/y) < 1/2,
+which holds since the maximum value of y �→ y log(1/y) is 1/e < 1/2.
+38
+
+Next, by applying the union bound over s ∈ [T1 + θτ : T1 + τ] in the above result, we conclude that
+P ∞
+�
+∃s ∈ [T1 + θτ : T1 + τ] : KL(p∥ˆqs−1) > ε2
+200
+�
+≤
+T1+τ
+�
+s=T1+θτ
+A′ exp
+�
+−s
+ε2
+A′ log(1/ε)
+�
+= A′ exp
+�
+−(T1 + θτ)
+ε2
+A′ log(1/ε)
+�
+·
+1
+1 − exp (−ε2/(A′ log(1/ε)),
+≤ A′2 log(1/ε)
+ε2
+exp
+�
+−θτ
+ε2
+A′ log(1/ε)
+�
+.
+where the equality sums over the geometric series, and the final inequality uses that T1 ≥ 0 and that for
+u < 1, 1/(1 − e−u) ≤ 2
+u.
+Next, observe that since for any x ∈ [−1, 1]d,
+1
+B2 ≤
+p(x)
+ˆqt−1(x) ≤ B2, we have the bound | log(p(Xt)/ˆqt−1(Xt))| ≤
+2 log B. Therefore, the Azuma-Hoeffding inequality is applicable, and yields that for every τ ≥ 1, δ > 0
+P ∞
+�
+T1+τ
+�
+s=T1+θτ
+log
+p(Xs)
+ˆqs−1(Xs) >
+T1+τ
+�
+s=T1+θτ
+KL(p∥ˆqs−1) + (τ − θτ)δ
+�
+≤ exp
+�
+−(τ − θτ)δ2/8 log2 B
+�
+.
+We proceed by setting δ = ε2/200 in the above, and applying the union bound, to conclude that there
+exists a constant C such that
+P ∞
+�
+T1+τ
+�
+s=T1+θτ
+log
+p(Xs)
+ˆqs−1(Xs) > (1 − θ)τε2
+100
+�
+≤ exp
+�
+−(1 − θ)τε4
+C log2 B
+�
++ C log(1/ε)
+ε2
+exp
+�
+−θτ
+ε2
+C log(1/ε)
+�
+.
+(6)
+Let us call the right hand side of (6) π(τ, θ). By the definition of ρt, and the boundedness of log
+p(x)
+ˆqs−1(x) for
+every s, it follows that with probability at least 1 − π(τ, θ),
+ρT1+τ(E ; p) =
+T1+τ
+�
+s=1
+log
+p(Xs)
+ˆqs−1(Xs)
+≤ 2(T1 + θτ) log B + (1 − θ)τε2
+100
+.
+So long as we can choose θ, M such that the upper bound above is smaller than (τ +T1)ε2/25, the inequality
+(6) will limit the probability that ρT1+τ > ε2(T1 + τ)/25, which is precisely our goal. But observe that this
+indeed occurs if (θ + 1/M) ≤ 3ε2/(200 log B), since in such a case
+2(T1 + θτ) log B + (1 − θ)τε2
+100
+≤ τ
+�
+2(1/M + θ) log B + ε2
+100
+�
+≤ τ
+�
+3ε2
+200 log B · 2 log B + ε2
+100
+���
+= τε2
+25 ≤ (T1 + τ)ε2
+25
+.
+So, we may set θ = min(1/2, ε2/(100 log B)) and M = max(1, (200 log B)/ε2), and conclude that for any
+τ ≥ MT1, it holds that
+P ∞(ρT1+τ(E ; p)/tε2 > 1/25) ≤ π(τ),
+where, for a constant C′,
+π(τ) = exp
+�
+−
+τε4
+C′ log2 B
+�
++ C′ log(1/ε)
+ε2
+exp
+�
+−
+τε2
+C′ log(1/ε) · log B
+�
+.
+39
+
+Note that in the terminology of Theorem 14, πt = π(t − T1) for t ≥ (M + 1)T1. Of course we can always
+provide the trivial bound πt ≤ 1 for t < (M + 1)T1. It remains to compute the resulting bound on expected
+rejection time. To this end, observe by summing the appropriate geometric series that
+�
+t≥1
+πt ≤ (M + 1)T1 +
+∞
+�
+τ=MT1
+π(τ)
+≤ (M + 1)T1 +
+1
+1 − exp
+�
+−ε4/C′ log2 B
+� +
+C′ log(1/ε)
+ε2(1 − exp (−ε2/C′ log(1/ε) · log B)
+≤ O
+� 1
+ε2
+�
+T1 + �O
+� 1
+ε4
+�
+,
+where the O bounds are as ε → 0, and we have hidden the dependence on B and log(1/ε). But, since in
+Lemma 24, ζt = �O(t−1/2(d+2)), and since T1 is the first time that ζ2
+t log2(1/ζt) ≤ Aε2/200, we may conclude
+that T1 = �O(ε−2(d+2)). The claim follows upon noticing that O(ε−2) · T1 = �O(ε−2(d+3)), and recalling that
+ε = dH(p, L).
+We conclude with a brief proof of Corollary 11 that exploits the bounds developed in the argument above.
+Proof of Corollary 11. It suffices to argue that using the estimators in the proof of Corollary 15, for any
+P ∈ DBox,Lip,B,
+P ∞
+�
+lim sup ρt(E ; p)
+td2
+H(p, L) ≤ 1
+25
+�
+= 1.
+This follows since for each t,
+P ∞
+� ρt(E ; p)
+td2
+H(p, L) > 1
+25
+�
+≤ πt,
+and � πt < ∞, which yields precisely the above relation by the Borel-Cantelli Lemma.
+40
+