Strong Law Of Large Numbers Research Articles

Stochastic flocking dynamics of the Cucker–Smale model driven by Lévy noise

Lévy noise is a broader type of white noise, which describes discontinuous noise and is more closely to simulate the realistic environment. In this paper, we introduce Lévy noise into the Cucker–Smale model to investigate its effect on the flocking dynamics of the system. The well-posedness of the system is guaranteed by defining an appropriate stopping time and constructing a Lyapunov function. Through the strong law of large numbers for martingales, a sufficient framework of the strong stochastic flocking is obtained without the strict assumption that the communication rate function has a positive lower bound due to the introduction of Lévy noise. We find that Lévy noise can accelerate flocking and become good noise in some cases. Finally, several numerical simulations are presented to validate our results and further observe effects of Lévy noise.

Strong laws of large numbers for weighted sums of 𝑑-dimensional arrays of random variables and applications to marked point processes

The aim of this paper is twofold. Firstly, we give equivalent conditions for the convergence of each of three series in the Kolmogorov-type three-series theorem for random variables with multidimensional indices. Our results extend the results of Sung (2011) for multi-indexed sums. Secondly, by using these results, we develop almost sure convergence for weighted sums of d d -dimensional arrays of random variables. We obtain the Jajte type strong law of large number for these arrays of random variables. As an application, the almost sure asymptotic behavior for sums generated by marked point processes is investigated.

Fluctuations of the Occupation Density for a Parking Process

Consider the following simple parking process on Λn:={-n,…,n}d,d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n:= \\{-n, \\ldots , n\\}^d,d\\ge 1$$\\end{document}: at each step, a site i is chosen at random in Λn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n$$\\end{document} and if i and all its nearest neighbor sites are empty, i is occupied. Once occupied, a site remains so forever. The process continues until all sites in Λn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n$$\\end{document} are either occupied or have at least one of their nearest neighbors occupied. The final configuration (occupancy) of Λn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n$$\\end{document} is called the jamming limit and is denoted by XΛn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{\\Lambda _n}$$\\end{document}. Ritchie (J Stat Phys 122:381–398, 2006) constructed a stationary random field on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}^d$$\\end{document} obtained as a (thermodynamic) limit of the XΛn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{\\Lambda _n}$$\\end{document}’s as n tends to infinity. As a consequence of his construction, he proved a strong law of large numbers for the proportion of occupied sites in the box Λn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _n$$\\end{document} for the random field X. Here we prove the central limit theorem, the law of iterated logarithm, and a gaussian concentration inequality for the same statistics. A particular attention will be given to the case d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}, in which we also obtain new asymptotic properties for the sequence XΛn,n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{\\Lambda _n},n\\ge 1$$\\end{document}.

Measure-Theoretic Analysis of Stochastic Competence Sets and Dynamic Shapley Values in Banach Spaces

We develop a measure-theoretic framework for dynamic Shapley values in Banach spaces, extending classical cooperative game theory to continuous-time, infinite-dimensional settings. We prove the existence and uniqueness of strong solutions to stochastic differential equations modeling competence evolution in Banach spaces, establishing sample path continuity and moment estimates. The dynamic Shapley value is rigorously defined as a càdlàg stochastic process with an axiomatic characterization. We derive a martingale representation for this process and establish its asymptotic properties, including a strong law of large numbers and a functional central limit theorem under α-mixing conditions. This framework provides a rigorous basis for analyzing dynamic value attribution in abstract spaces, with potential applications to economic and game-theoretic models.

The convergence properties for randomly weighted sums of widely negative dependent random variables under sub-linear expectations with related statistical applications

In this paper, we study the complete convergence and complete moment convergence for randomly weighted sums of arrays of rowwise widely negative dependent random variables in sub-linear expectation space under some appropriate conditions, which extend and improve the corresponding ones in classical probability space to the case of sub-linear expectation space. And we obtain a strong law of large numbers for the randomly weighted sums of arrays of rowwise widely negative dependent random variables. As applications of our main results, we not only present a result on the complete consistency for the weighted estimator in a nonparametric regression model, but also obtain the complete consistency for the least squares estimators in errors-in-variables regression models based on widely negative dependent errors under sub-linear expectations. We perform some numerical simulations to verify the validity of the theoretical results and a real example is analysed for illustration.

Strong laws for weighted sums of widely orthant dependent random variables and applications

Abstract In this study, the strong law of large numbers and the convergence rate for weighted sums of non-identically distributed widely orthant dependent random variables are established. As applications, the strong consistency for weighted estimator in nonparametric regression model and the rate of strong consistency for least-squares estimator in multiple linear regression model are obtained. Some numerical simulations are also provided to verify the validity of the theoretical results.

On complete convergence for weighted sums of m-widely acceptable random variables under sub-linear expectations and its statistical applications

. In this article, by means of Rosenthal-type inequality, we study the complete convergence and strong law of large numbers for weighted sums of m-widely accpetable random variables under sub-linear expectations, which extend some existing results in classical probability space. As applications, we investigate the complete consistency and strong consistency of weighted estimators in nonparametric regression models under sub-linear expectations. In addition, we provide some simulation studies to verify the finite sample performance of the theoretical results.

Trajectory fitting estimation for integrated Ornstein-Uhlenbeck process driven by Lévy process

In this article, we investigate the trajectory fitting estimation for the integrated Ornstein-Uhlenbeck process driven by the Lévy process based on continuous observations. The strong consistency and asymptotic distributions of the estimator are obtained by the strong law of large numbers and the inner clock property of the α-stable stochastic integral.

Divergences based Bayesian inference with censored data

. In this work, we deal with some Bayesian inference problems in the presence of right censored data. First, we propose a dual ϕ−divergence Bayes type estimators for parametric models and we establish their asymptotic normality. To establish this result, we need a uniform strong law of large numbers that we prove as well. Then, we consider the problem of prior distributions construction for model selection using ϕ−divergences. Finally, we consider the problem of the predictive density estimation on the basis of ϕ−divergences. We apply an expansion result of the generalized Bayesian predictive density on two parametric models widely used in survival analysis, namely the Weibull and the inverse Weibull model, under right censoring. We also check the performances of our proposed methods through simulations and real data applications. The results of these studies show that our proposed dual ϕ−divergence Bayes type estimators are more robust than other Bayesian estimators. Moreover, the generalized Bayesian predictive density performs better than the classical estimative density especially for the inverse Weibull model.

Almost sure exponential stability and stochastic stabilization of discrete-time stochastic systems with impulses

This paper considers the almost sure exponential stability and stochastic stabilization problems of discrete-time stochastic systems (DTSSs) with impulsive effects, where the average impulsive interval is taken into account. By using the average impulsive interval approach and the strong law of large numbers, we not only establish the criteria for almost sure exponential stability of general nonlinear discrete-time impulsive stochastic systems (DTISSs) but also design an exact method of a stochastic perturbation to stabilize a given unstable impulsive discrete-time systems. Adopting the average impulsive interval approach and the strong law of large numbers, we established a criterion for almost sure exponential stability of general nonlinear DTISSs. Furthermore, a method of stochastic perturbation has been developed to stabilize an unstable impulsive discrete-time system. Finally, two simulation examples demonstrate the effectiveness of the derived results.

On Asymptotic Equipartition Property for Stationary Process of Moving Averages

Let {Xn}n∈Z be a stationary process with values in a finite set. In this paper, we present a moving average version of the Shannon–McMillan–Breiman theorem; this generalize the corresponding classical results. A sandwich argument reduced the proof to direct applications of the moving strong law of large numbers. The result generalizes the work by Algoet et. al., while relying on a similar sandwich method. It is worth noting that, in some kind of significance, the indices an and ϕ(n) are symmetrical , i.e., for any integer n, if the growth rate of (an)n∈Z is slow enough, all conclusions in this article still hold true.

From law of the iterated logarithm to Zolotarev distance for supercritical branching processes in random environment

Consider (Zn)n⩾0 a supercritical branching process in an independent and identically distributed environment. Based on some recent development in martingale limit theory, we established law of the iterated logarithm, strong law of large numbers, invariance principle and optimal convergence rate in the central limit theorem under Zolotarev and Wasserstein distances of order p∈(0,2] for the process (logZn)n⩾0.

Complete Convergence for Weighted Sums of Pairwise Negative Quadrant Dependent Random Variables

In this work, we developed Jajte’s technique of the strong law of large numbers to the complete convergence for randomly weighted sums of pairwise negative quadrant dependent random variables.

AN APPROACH OF MEAN-SETS THEORY FOR NEGATIVELY CURVED CONVEX COMBINATION POLISH METRIC SPACES

The choice of a convenient approach to be used is one important issue when attempting to develop methods or obtain results in the setting of Probability on general metric spaces. In this paper, we extend the mean-sets probabilistic approach formally introduced by Mosina on locally finite graphs, and hence (via Cayley graphs) on finitely generated groups, to the field of Negatively Curved Convex Combination Polish (NCCCP) metric spaces. We construct an appropriated Vertex-Weighted Metric (VWM) graph in the framework of this class of geometrical structures. We define a function called convexification function on the direct product of $ n $ copies of the vertex-set of this graph (for a given fixed integer $ n\geq 2$), using the natural convexification operator of the metric space concerned. This function is then used to construct a weighted mean-set that generalizes the notion of convex combination (CC) mean in the sense of Ter{\'a}n and Molchanov, the mean-set concept according to Mosina and the ordinary notion of $ k $-means ($k\geq 2$) of independent identically distributed (i.i.d.) random elements of the metric space. Two numerical examples are given for the cases when the metric space $ X= [0,1]$ and $ X=\mathbf{R}^{2} $. Moreover, an analogue of the Strong Law of Large Numbers (SLLN), the consistency problem and the Chebyshev's inequality for NCCCP spaces are established.

Functional central limit theorem and Marcinkiewicz strong law of large numbers for Hilbert-valued U-statistics of absolutely regular data

Functional central limit theorem and Marcinkiewicz strong law of large numbers for Hilbert-valued U-statistics of absolutely regular data

Strong law of large numbers for double sequences of pairwise M-dependent real and multivalued random variables

Strong law of large numbers for double sequences of pairwise M-dependent real and multivalued random variables

Distributions of 4-subtree patterns for uniform random unrooted phylogenetic trees

Tree shape statistics based on peripheral structures have been utilized to study evolutionary mechanisms and inference methods. Partially motivated by a recent study by Pouryahya and Sankoff on modeling the accumulation of subgenomes in the evolution of polyploids, we present the distribution of subtree patterns with four or fewer leaves for the unrooted Proportional to Distinguishable Arrangements (PDA) model. We derive a recursive formula for computing the joint distributions, as well as a Strong Law of Large Numbers and a Central Limit Theorem for the joint distributions. This enables us to confirm several conjectures proposed by Pouryahya and Sankoff, as well as provide some theoretical insights into their observations. Based on their empirical datasets, we demonstrate that the statistical test based on the joint distribution could be more sensitive than those based on one individual subtree pattern to detect the existence of evolutionary forces such as whole genome duplication.

Conditional Strong Law of Large Numbers under G-Expectations

In this paper, we investigate two types of the conditional strong law of large numbers with a new notion of conditionally independent random variables under G-expectation which are related to the symmetry G-function. Our limit theorem demonstrates that the cluster points of empirical averages fall within the bounds of the lower and upper conditional expectations with lower probability one. Moreover, for conditionally independent random variables with identical conditional distributions, we show the existence of two cluster points of empirical averages that correspond to the essential minimum and essential maximum expectations, respectively, with G-capacity one.

Jajte-type strong limit theorem for pairwise negatively quadrant dependent random variables

The authors study the strong limit theorems for pairwise negatively quadrant dependent random variables, and present a new Jajte-type theorem on the complete convergence and the strong laws of large numbers. The results obtained in this article improve some corresponding theorems in the existing literature.

Product disintegrations: A law of large numbers via conditional independence

We introduce the concept of a product disintegration, which generalizes exchangeability by allowing one to render conditional independence in terms of a suitable hidden sequence of random probabilities. We prove that, given a sequence of random elements in a compact metric space, a strong law of large numbers (SLLN) holds for observables of this process if and only if the process admits a product disintegration that itself satisfies a type of SLLN. As an application, we provide a characterization of the SLLN for identically distributed (not necessarily independent) Bernoulli sequences.