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.     

Weak convergence of random sums and maximum random sums under nonrandom norming

Necessary and sufficient conditions are obtained for the weak convergence of random sums and maximum random sums of independent identically distributed random variables. Limit distributions for these sums are described. The indices are not assumed to be independent of the summands. Supported in part by Grants NFW000 and NFW300 from the International Science Foundation and the Russian Government and by Grant No. 93-011-1446 from the Russian Foundation for Fundamental Research. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

行为负相依随机变量阵列加权和的完全收敛性

本文研究了行为NOD随机变量阵列加权和的完全收敛性.运用NOD随机变量列的矩不等式以及截尾的方法,得到了关于行为NOD随机变量阵列加权和的完全收敛性的充分条件.利用获得的充分条件,推广了Baek(2008)关于行为NA随机变量阵列加权和的完全收敛性的结论,得到了比吴群英(2012)更为一般的结果.     

On the convergence in law of almost all sums of independent random variables

The paper deals with sums of independent and identically distributed random variables defined on some probability space which are multiplied by random coefficients. These coefficients are the values of independent random variables defined on another probability space. We obtain conditions for the weak convergence of weighted sums, for almost all coefficients, to some infinitely divisible distribution. The limit distribution for these sums is found. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-16099). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

随机变量序列乘积和的矩完全收敛性

利用王岳宝等将乘积和转化为部分和的乘积之和的方法,研究了随机变量序列乘积和的矩完全收敛性,获得了乘积和矩完全收敛的充分条件.     

Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables

In this paper,the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables are investigated.Some sufficient conditions for the convergence are provided.In addition,the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables is obtained.The results obtained in the article extend the corresponding ones for independent random variables and some dependent random variables.     

Convergence rates for weighted sums in noncommutative probability space

We study convergence rates for weighted sums of pairwise independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. As applications, we first study convergence rates for weighted sums of random variables in the noncommutative Lorentz space, and second we study convergence rates for weighted sums of probability measures with respect to the free additive convolution.     

B值随机元和的完全收敛性

在B值随机元随机有界于一非负随机变量的情况下．讨论了B值独立随机元序列非随机足标和的完全收敛性，作为应用，得到了随机足标和完全收敛性的相应结果，将[5]中的一些结果推广到B值独立随机无情形，同时使[3]（d=1），[4]，[6]的相应结果成为特例．     

Bounds for some general sums of random variables

Rogers and Shi (1995) have used the technique of conditional expectations to derive approximations for the distribution of a sum of lognormals. In this paper we extend their results to more general sums of random variables. In particular we study sums of functions of dependent random variables that are multivariate normally distributed and also derive results for sums of functions of dependent random variables from the additive exponential dispersion family. The usefulness of our results for practical applications is also discussed.     

非同分布NA序列的完全收敛性

讨论了非同分布NA序列部分和与随机足标部分和的完全收敛性，推广了于浩在1989年得到的关于独立随机变量序列的一些结果。     

宽相依随机变量加权和的强极限定理

Strong limit theorems are established for weighted sums of widely orthant dependent（WOD） random variables. As corollaries, the strong limit theorems for weighted sums of extended negatively orthant dependent（ENOD） random variables are also obtained, which extend and improve the related known works in the literature.     

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??In this paper, the complete moment convergence for weighted sums of-mixing random variable series are investigated. By using Rosenthal type inequality, we obtain complete moment convergence theorems for weighted sums of-mixing random variable series, which generalize and improve the corresponding results.     

A criterion of convergence of nonrandomly centered random sums of independent identically distributed random variables

Necessary and sufficient conditions are presented for the weak convergence of random sums of independent identically distributed random variables in the double array scheme. As corollaries, two criteria of the normal convergence of random sums are given. Supported by the Russian Foundation for Fundamental Research (grant No. 96-011-01919). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Precise Large Deviations for Long-Tailed Distributions

In this paper, we investigate the precise large deviations for sums of independent identically distributed random variables with heavy-tailed distributions. We prove asymptotic relations for non-random sums and for random sums of random variables with long-tailed distributions. We apply the results on two useful counting processes, namely, renewal and compound-renewal processes.     

随机变量序列加权和的强收敛性

本文讨论了一般随机变量序列加权和的强收敛性．作为推论，得到一类鞅差序列加权和的收敛定理和若干经典的独立随机变量序列的强大数定律；已有的若干结论是本文结果的特例．     

长尾宽上限相依的不同分布的随机变量和的精确大偏差

研究了服从长尾分布族上的随机变量和的精确大偏差问题,其中假设代表索赔额的随机变量序列是一列宽上限相依的、不同分布的随机变量序列。在给定一些假设条件下,得到了部分和与随机和的两种一致渐近结论。     

行为ND随机变量阵列加权和的完全收敛性

在非同分布的情况下,给出了行为ND随机变量阵列加权和的完全收敛性的充分条件,所得结果部分地推广了独立随机变量和NA随机变量的相应结果.作为其应用,获得了ND随机变量序列加权和的Marcinkiewicz-Zygmund型强大数定律.     

行为混合阵列加权和的收敛性

本文研究了行为混合阵列加权和的收敛性.利用混合序列的Rosenthal型最大值不等式,讨论了混合阵列加权和的L1收敛性,依概率收敛性,几乎处处收敛性,及完全收敛性之间的等价关系,推广了行独立随机变量阵列相应的结果.     

Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails

This paper studies the moderate deviations of real-valued extended negatively dependent (END) random variables with consistently varying tails. The moderate deviations of partial sums are first given. The results are then used to establish the necessary and sufficient conditions for the moderate deviations of random sums under certain circumstances.     

On the strong law of large numbers for sequences of dependent random variables

New sufficient conditions for the applicability of the strong law of large numbers are established for sequences of random variables without the independence conditions. Results on strong stability of sums of dependent random variables are also obtained. No particular type of dependence between random variables of a sequence is assumed. Only conditions related to moments of random variables and their sums are used. It is shown that the results obtained are unimprovable in certain sense. These results are generalizations of some results of N. Etemadi proved under more restrictive conditions.