General theorems on rates of convergence in distribution of random variables I. General limit theorems

Let (X_n)_n?$\text{N}$ be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions F_Xn, and S_n = Σ_i=1ⁿX_i. The authors present limit theorems together with convergence rates for the normalized sums ?(n)S_n, where ?: $\text{N}$ → $\text{R}$⁺, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝_$\text{R}$f(x) d[F_?(n)Sn(x) ? F_X(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.     

Rate of convergence bounds in local limit theorems for sums of a random number of independent random variables

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 48–54, 1986.     

On central limit theorems for shrunken random variables

We discuss Central Limit Theorems and absence of limiting distributions for shrunken random variables.

     

AANA随机变量序列的中心极限定理

研究了渐近几乎负相依(简称为AANA)随机变量序列的渐近正态问题.在非常一般的条件下,得到了AANA序列的中心极限定理,推广了负相依(简称为NA)、独立随机变量序列的相应结论.     

Central limit theorems for generalized set-valued random variables

We give central limit theorems for generalized set-valued random variables whose level sets are compact both in or in a Banach space under milder conditions than those obtained recently by the latter two authors.     

Some limit theorems for negatively associated random variables

Let {X _n , n ≥ 1} be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the L ^p -convergence theorem and Marcinkiewicz–Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables.     

Almost sure limit theorems for random sums of multiindex random variables

In this paper, we obtain an almost sure functional limit theorem for random sums of multiindex random variables.     

Conditional limit theorems for conditionally negatively associated random variables

From the ordinary notion of negative association for a sequence of random variables, a new concept called conditional negative association is introduced. The relation between negative association and conditional negative association is answered, that is, the negative association does not imply the conditional negative association, and vice versa. The basic properties of conditional negative association are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some Rosenthal type inequalities for maximum partial sums of such sequences of random variables are derived, which extend the corresponding results for negatively associated random variables. As applications of these inequalities, some conditional mean convergence theorems, conditionally complete convergence results and a conditional central limit theorem stated in terms of conditional characteristic functions are established. In addition, some lemmas in the context are of independent interest.     

On strong limit theorems concerning delayed sums of a random sequence

Let (ξ _n ) _n∈N be a sequence of arbitrarily dependent random variables. In this paper, a generalized strong limit theorem of the delayed average of (ξ _n ) _n∈N is investigated, then some limit theorems for arbitrary information sources follow. As corollaries, some known results are generalized.     

Conditional versions of limit theorems for conditionally associated random variables

Relation between association and conditional association is answered, several examples show that the association of random variables does not imply the conditional association, and vice versa. Several fundamental properties of conditional associated random variables are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional Hájek-Rényi type inequalities, a conditional strong law of large numbers and a conditional central limit theorem stated in terms of conditional characteristic functions are established, which are conditional versions of the earlier results for associated random variables, respectively. In addition, some lemmas in the context are of independent interest.     

矩条件下的任意随机变量序列的一类强极限定理

本文利用随机变量的截尾方法和条件三级数定理,研究了任意随机变量序列在矩条件下的一类强极限定理,改进了与此相应的一些结果的条件.     

One-sided limit theorems for sums of multidimensional indexed random variables

Let {X,X _n,nZ ₊ ^d } be a sequence of independent and identically distributed random variables and {a _n ,n Z ₊ ^d } be a sequence of constants. We examine the almost sure limiting behavior of weighted partial sums of the form _|n|N a _n X _n . Suppose further that eitherEX=0 orE|X|=. In most situations these normalized partial sums fail to have a limit, no matter which normalizing sequence we choose. Thus, the investigation lends itself to the study of the limit inferior and limit superior of these sequences. On the way to proving results of this type we first establish several weak laws. These weak laws prove to be of great value in establishing generalized laws of the iterated logarithm.     

Almost sure versions of limit theorems for random sums of multiindex random variables

In this paper we obtain an almost sure version of a limit theorem for random sums of multiindex random variables that belong to the domain of attraction of a p-stable law.     

Some new limit theorems for vector space- and group-valued random variables

It is shown that weakly convergent sums (products) of normalized i.i.d. random variables with values in a finite-dimensional vector space or in a group are mixing in the sense of A. Rényi, and limit theorems for random sums with nonindependent indices are obtained. A new version of H. Robbins' limit theorem for random sums is presented. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló Hungary, 1997, Part III.     

Some generalized limit theorems concerning delayed sums of random sequence

For a sequence of arbitrarily dependent m-valued random variables (Xn) n∈N , the generalized strong limit theorem of the delayed average is investigated. In our proof, we improved the method proposed by Liu [6] . As an application, we also studied some limit properties of delayed average for inhomogeneous Markov chains.