Law Of The Iterated Logarithm Research Articles

Diophantine conditions in the law of the iterated logarithm for lacunary systems

AbstractIt is a classical observation that lacunary function systems exhibit many properties which are typical for systems of independent random variables. However, it had already been observed by Erdős and Fortet in the 1950s that probability theory’s limit theorems may fail for lacunary sums $$\sum f(n_k x)$$ ∑ f ( n k x ) if the sequence $$(n_k)_{k \ge 1}$$ ( n k ) k ≥ 1 has a strong arithmetic “structure”. The presence of such structure can be assessed in terms of the number of solutions $$k,\ell $$ k , ℓ of two-term linear Diophantine equations $$a n_k - b n_\ell = c$$ a n k - b n ℓ = c . As the first author proved with Berkes in 2010, saving an (arbitrarily small) unbounded factor for the number of solutions of such equations compared to the trivial upper bound, rules out pathological situations as in the Erdős–Fortet example, and guarantees that $$\sum f(n_k x)$$ ∑ f ( n k x ) satisfies the central limit theorem (CLT) in a form which is in accordance with true independence. In contrast, as shown by the first author, for the law of the iterated logarithm (LIL) the Diophantine condition which suffices to ensure “truly independent” behavior requires saving this factor of logarithmic order. In the present paper we show that, rather surprisingly, saving such a logarithmic factor is actually the optimal condition in the LIL case. This result reveals the remarkable fact that the arithmetic condition required of $$(n_k)_{k \ge 1}$$ ( n k ) k ≥ 1 to ensure that $$\sum f(n_k x)$$ ∑ f ( n k x ) shows “truly random” behavior is a different one at the level of the CLT than it is at the level of the LIL: the LIL requires a stronger arithmetic condition than the CLT does.

The Law of the Iterated Logarithm for Lp-Norms of Kernel Estimators of Cumulative Distribution Functions

In this paper, we consider the strong convergence of Lp-norms (p≥1) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the Lp-norms of empirical processes is extended to the kernel estimator of the CDF.

A law of the iterated logarithm for small counts in Karlin’s occupancy scheme

In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1,2,\dots $ , with probability ${p_{k}}$ of hitting the box k. For $j,n\in \mathbb{N}$, denote by ${\mathcal{K}_{j}^{\ast }}(n)$ the number of boxes containing exactly j balls provided that n balls have been thrown. Small counts are the variables ${\mathcal{K}_{j}^{\ast }}(n)$, with j fixed. The main result is a law of the iterated logarithm (LIL) for the small counts as the number of balls thrown becomes large. Its proof exploits a Poissonization technique and is based on a new LIL for infinite sums of independent indicators ${\textstyle\sum _{k\ge 1}}{1_{{A_{k}}(t)}}$ as $t\to \infty $, where the family of events ${({A_{k}}(t))_{t\ge 0}}$ is not necessarily monotone in t. The latter LIL is an extension of a LIL obtained recently by Buraczewski, Iksanov and Kotelnikova (2023+) in the situation when ${({A_{k}}(t))_{t\ge 0}}$ forms a nondecreasing family of events.

Insight into Spatially Colored Stochastic Heat Equation: Temporal Fractal Nature of the Solution

In this paper, the solution to a spatially colored stochastic heat equation (SHE) is studied. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time has exact, dimension-dependent, global continuity moduli, and laws of the iterated logarithm (LILs). It is obtained that the set of fast points at which LILs fail in this process, and occur infinitely often, is a random fractal, the size of which is evaluated by its Hausdorff dimension. These points of this process are everywhere dense with the power of the continuum almost surely, and their hitting probabilities are determined by the packing dimension dimP(E) of the target set E.

One-Sided Version of Law of the Iterated Logarithm for Summations of Signum Functions

The law of the iterated logarithm (LIL), which describes the rate of convergence for a convergent lacunary series, was established by R. Salem and A. Zygmund. This rate is determined based on the variance-like term of the remainder after n terms of the series. In this article, we investigate a comparable one-sided LIL for sums of signum functions, which also relies on the remainder after n terms.

Law of Large Numbers, Central Limit Theorem, and Law of the Iterated Logarithm for Bernoulli Uncertain Sequence

In order to describe human uncertainty more precisely, Baoding Liu established uncertainty theory. Thus far, uncertainty theory has been successfully applied to uncertain finance, uncertain programming, uncertain control, etc. It is well known that the limit theorems represented by law of large numbers (LLN), central limit theorem (CLT), and law of the iterated logarithm (LIL) play a critical role in probability theory. For uncertain variables, basic and important research is also to obtain the relevant limit theorems. However, up to now, there has been no research on these limit theorems for uncertain variables. The main results to emerge from this paper are a strong law of large numbers (SLLN), a weak law of large numbers (WLLN), a CLT, and an LIL for Bernoulli uncertain sequence. For studying these theorems, we first propose an assumption, which can be regarded as a generalization of the duality axiom for uncertain measure in the case that the uncertainty space can be finitely partitioned. Additionally, several new notions such as weakly dependent, Bernoulli uncertain sequence, and continuity from below or continuity from above of uncertain measure are introduced. As far as we know, this is the first study of the LLN, the CLT, and the LIL for uncertain variables. All the theorems proved in this paper can be applied to uncertain variables with symmetric or asymmetric distributions. In particular, the limit of uncertain variables is symmetric in (c) of the third theorem, and the asymptotic distribution of uncertain variables in the fifth theorem is symmetrical.

A functional law of the iterated logarithm for weakly hypoelliptic diffusions at time zero

We study the almost sure behavior of solutions of stochastic differential equations (SDEs) as time goes to zero. Our main general result establishes a functional law of the iterated logarithm (LIL) that applies in the setting of SDEs with degenerate noise satisfying the weak Hörmander condition but not the strong Hörmander condition. That is, SDEs in which the drift terms must be used in order to conclude hypoellipticity. As a corollary of this result, we obtain the almost sure behavior as time goes to zero of a given direction in the equation, even if noise is not present explicitly in that direction. The techniques used to prove the main results are based on large deviations applied to a non-trivial rescaling of the original system. In concrete examples, we show how to find the proper rescaling to obtain the functional LIL. Furthermore, we apply the main results to the problem of identifying regular points for hypoelliptic diffusions. Consequently, we obtain a control-theoretic criteria for a given point to be regular for the process.

Exact Uniform Modulus of Continuity and Chung’s LIL for the Generalized Fractional Brownian Motion

The generalized fractional Brownian motion (GFBM) \(X:=\{X(t)\}_{t\ge 0}\) with parameters \(\gamma \in [0, 1)\) and \(\alpha \in \left( -\frac{1}{2}+\frac{\gamma }{2}, \, \frac{1}{2}+\frac{\gamma }{2} \right) \) is a centered Gaussian H-self-similar process introduced by Pang and Taqqu (2019) as the scaling limit of power-law shot noise processes, where \(H = \alpha -\frac{\gamma }{2}+\frac{1}{2} \in (0,1)\). When \(\gamma = 0\), X is the ordinary fractional Brownian motion. When \(\gamma \in (0, 1)\), GFBM X does not have stationary increments, and its sample path properties such as Hölder continuity, path differentiability/non-differentiability, and the functional law of the iterated logarithm (LIL) have been investigated recently by Ichiba et al. (J Theoret Probab 10.1007/s10959-020-01066-1, 2021). They mainly focused on sample path properties that are described in terms of the self-similarity index H (e.g., LILs at infinity or at the origin). In this paper, we further study the sample path properties of GFBM X and establish the exact uniform modulus of continuity, small ball probabilities, and Chung’s laws of iterated logarithm at any fixed point \(t > 0\). Our results show that the local regularity properties away from the origin and fractal properties of GFBM X are determined by the index \(\alpha +\frac{1}{2}\) instead of the self-similarity index H. This is in contrast with the properties of ordinary fractional Brownian motion whose local and asymptotic properties are determined by the single index H.

Uniform Limit Theorems under length-biased sampling and type I censoring

In recent years, in view of theory of empirical processes, authors have become more interested in the uniform analogue of the three fundamental theorems: the uniform law of large numbers of Glivenko-Cantelli type, the uniform central limit theorem for Donsker type and the functional law of the iterated logarithm (LIL). In this paper, under the bracketing entropy conditions, the uniform law of large numbers, uniform central limit theorem and the uniform LIL of Strassen type have been investigated in the case of length-biased and type I censoring.

On the law of iterated logarithm for extreme queue length in an open queueing network

The purpose of this research in the field of the open queueing network is to prove the Law of the Iterated Logarithm (LIL) for the extreme value of the queue length of customers in an open queueing network. LIL is proved for the extreme values of the queue length of customers the important probability characteristic of the queueing system under conditions of heavy traffic. Also, we present for extreme queue length of jobs Probability Laws ((theorems on the LIL, Fluid Limits Theorem (FLT) and Diffusion Limit Theorem (DLT)) in various conditions of traffic and simulating an open queueing network in Appendix A and Appendix B.

Time-uniform, nonparametric, nonasymptotic confidence sequences

A confidence sequence is a sequence of confidence intervals that is uniformly valid over an unbounded time horizon. Our work develops confidence sequences whose widths go to zero, with nonasymptotic coverage guarantees under nonparametric conditions. We draw connections between the Cram\'er-Chernoff method for exponential concentration, the law of the iterated logarithm (LIL), and the sequential probability ratio test -- our confidence sequences are time-uniform extensions of the first; provide tight, nonasymptotic characterizations of the second; and generalize the third to nonparametric settings, including sub-Gaussian and Bernstein conditions, self-normalized processes, and matrix martingales. We illustrate the generality of our proof techniques by deriving an empirical-Bernstein bound growing at a LIL rate, as well as a novel upper LIL for the maximum eigenvalue of a sum of random matrices. Finally, we apply our methods to covariance matrix estimation and to estimation of sample average treatment effect under the Neyman-Rubin potential outcomes model.

Law of iterated logarithm and model selection consistency for generalized linear models with independent and dependent responses

We study the law of the iterated logarithm (LIL) for the maximum likelihood estimation of the parameters (as a convex optimization problem) in the generalized linear models with independent or weakly dependent (ρ-mixing) responses under mild conditions. The LIL is useful to derive the asymptotic bounds for the discrepancy between the empirical process of the log-likelihood function and the true log-likelihood. The strong consistency of some penalized likelihood-based model selection criteria can be shown as an application of the LIL. Under some regularity conditions, the model selection criterion will be helpful to select the simplest correct model almost surely when the penalty term increases with the model dimension, and the penalty term has an order higher than O(log log n) but lower than O(n). Simulation studies are implemented to verify the selection consistency of Bayesian information criterion.

On the law of the iterated logarithm for continued fractions with sequentially restricted partial quotients

We establish a law of the iterated logarithm (LIL) for the set of real numbers whose nth partial quotient is bigger than α n , where (α n ) is a sequence such that ∑1/α n is finite. This set is shown to have Hausdorff dimension 1/2 in many cases and the measure in LIL is absolutely continuous to the Hausdorff measure. The result is obtained as an application of a strong invariance principle for unbounded observables on the limit set of a sequential iterated function system.

A functional law of the iterated logarithm for multi-class queues with batch arrivals

A functional law of the iterated logarithm (LIL) and its corresponding LIL are established for a multiclass single-server queue with first come first served (FCFS) service discipline. The functional LIL and its LIL quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The functional LIL and LIL are established in three cases: underloaded, critically loaded and overloaded, for performance measures including the total workload, idle time, queue length, workload, busy time, departure and sojourn time processes. The proofs of the functional LIL and LIL are based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. Numerical examples are considered to provide insights on these limit results.

Limit theorems for locally stationary processes

We present limit theorems for locally stationary processes that have a one sided time-varying moving average representation. In particular, we prove a central limit theorem (CLT), a weak and a strong law of large numbers (WLLN, SLLN) and a law of the iterated logarithm (LIL) under mild assumptions using a time-varying Beveridge–Nelson decomposition.

The law of the iterated logarithm for a class of SPDEs

After establishing the moderate deviation principle by the classical Azencott method, we prove the Strassen’s compact law of the iterated logarithm (LIL) for a class of stochastic partial differential equations (SPDEs). As an application, we obtain this type of LIL for two population models known as super-Brownian motion and Fleming-Viot process. In addition, the classical LIL is shown for the class of SPDEs and the two population models.

Strong consistency and asymptotic normality for quantities related to the Benjamini–Hochberg false discovery rate procedure

The Benjamini–Hochberg (BH) false discovery rate (FDR) procedure enjoys widespread use in multiple testing scenarios when the cost of false positives is low and the number of simultaneous tests prohibits use of the Bonferroni procedure. The FDR and the average power are each expected values, of the false discovery proportion (FDP) and true positive proportion (TPP), respectively. Chi (2007), proved a law of the iterated logarithm (LIL) for the positive proportion (PP), FDP and TPP, and discusses a criticality phenomenon whereby a minimal FDR corresponding to the effect size is required for a nonzero rate of positive calls. While almost sure convergence follows as a corollary, no direct simple proof exists in the literature. We provide this result under the same set of conditions. Of greater consequence, we prove central limit results (CLT) for the PP, FPF and TPP under these weak conditions, providing full characterization of the limits.

On the Law of the Iterated Logarithm Without Assumptions About the Existence of Moments

On the Law of the Iterated Logarithm Without Assumptions About the Existence of Moments

On the analysis of the law of the iterated logarithm in open queueing networks

ABSTRACTThe modern queueing theory is one of the powerful tools for a quantitative and qualitative analysis of communication systems, computer networks, transportation systems, and many other technical systems. The paper is designated to the analysis of queueing systems, arising in the network theory and communications theory (called open queueing network). We have proved here the theorem on the law of the iterated logarithm (LIL) for the virtual waiting time of a job in an open queueing network under heavy and light traffic conditions. Also, the work presents a survey of papers for the virtual waiting time of a job in heavy traffic. Finally, we present an application of the proved theorems to the technical model from computer network practice.

Fine properties of fractional Brownian motions on Wiener space

We study several important fine properties for the family of fractional Brownian motions with Hurst parameter H under the (p,r)-capacity on classical Wiener space introduced by Malliavin. We regard fractional Brownian motions as Wiener functionals via the integral representation discovered by Decreusefond and Üstünel, and show non-differentiability, modulus of continuity, law of the iterated logarithm (LIL) and self-avoiding properties of fractional Brownian motion sample paths using Malliavin calculus as well as the tools developed in the previous work by Fukushima, Takeda and etc. for Brownian motion case.