Related papers: Multi-dimensional central limit theorems and laws …
In this paper, by using the representation theorem for sublinear expectations, we give a simple proof to obtain two inequalities about the sample mean for independent random vectors under sublinear expectations.
In this talk I first review at an elementary level a selection of central limit theorems, including some lesser known cases, for sums and maxima of uncorrelated and correlated random variables. I recall why several of them appear in…
Consider multiple sums $S_n$ on the $d$-dimensional integer grid,which are generated by i.i.d.\ random variables with a positive expectation. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional…
We provide a systematic approach to stable central limit theorems for d-dimensional martingale difference arrays and martingale difference sequences. The conditions imposed are straightforward extensions of the univariate case.
Ordinary differential equations obtained as limits of Markov processes appear in many settings. They may arise by scaling large systems, or by averaging rapidly fluctuating systems, or in systems involving multiple time-scales, by a…
We give a theory of sublinear expectations and martingales in discrete time. Without assuming the existence of a dominating probability measure, we derive the extensions of classical results on uniform integrability, optional stopping of…
The main purpose of this work is to introduce and analyse some generalizations of diverse superposition rules for first-order differential equations to the setting of second-order differential equations. As a result, we find a way to apply…
A central limit theorem for arrays of symmetric row-wise exchangeable random variables is presented. The result is valid for finite and infinite extendable and non-extendable sequences. Unlike most reported versions of the central limit…
We prove a central limit theorem for the length of the longest subsequence of a random permutation which follows one of a class of repeating patterns. This class includes every fixed pattern of ups and downs having at least one of each,…
Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have raised progressive interest recently. The purpose of this paper is to study the strong law of large numbers and the law of the…
This paper is concerned with Spearman's correlation matrices under large dimensional regime, in which the data dimension diverges to infinity proportionally with the sample size. We establish the central limit theorem for the linear…
In this paper, we establish a new law of large numbers with the rate of convergence for special partial sums in a probability space. The proof relies on nonlinear expectation theory, as the uncertainty of random variables in the special…
We take a unified approach to central limit theorems for a class of irreducible urn models with constant replacement matrix. Depending on the eigenvalue, we consider appropriate linear combinations of the number of balls of different…
We consider the multiparameter random simplicial complex on a vertex set $\{ 1,\dots,n \}$, which is parameterized by multiple connectivity probabilities. Our key results concern the topology of this complex of dimensions higher than the…
Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov…
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers…
Lower and upper bounds are explored for the uniform (Kolmogorov) and $L^2$-distances between the distributions of weighted sums of dependent summands and the normal law. The results are illustrated for several classes of random variables…
We investigate the rate of convergence in the central limit theorem for convex sets. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the…
We show how a central limit theorem for Poisson model random polygons implies a central limit theorem for uniform model random polygons. To prove this implication, it suffices to show that in the two models, the variables in question have…
Limit theorems of strong law of large numbers and central limit theorem types are obtained for the compositions of independent identically distributed random unitary channels.