On Infinitely Divisible Distributions on Locally Compact Abelian Groups

Our aim in this paper is to characterize some classes of infinitely divisible distributions on locally compact abelian groups. Firstly infinitely divisible distributions with no idempotent factor on locally compact abelian groups are characterized by means of limit distributions of sums of independent random variables. We introduce semi-selfdecomposable distributions on topological fields, and in case of totally disconnected fields we give a limit theorem for them. We also give a characterization of semistable laws on p-adic field and show that semistable processes are constructed as scaling limits of sums of i.i.d.     

Semi-selfdecomposable distributions and a new class of limit theorems

Self-decomposable distributions are given as limits of normalized sums of independent random variables. We define semi-selfdecomposable distributions as limits of subsequences of normalized sums. More generally, we introduce a way of making a new class of limiting distributions derived from a class of distributions by taking the limits through subsequences of normalized sums, and define the class of semi-selfdecomposable distributions and a decreasing sequence of subclasses of it. We give two kinds of necessary and sufficient conditions for distributions belonging to those classes, one is in terms of the decomposability of random variables and another is in terms of Lévy measures. Received: 1 May 1997 / Revised version: 5 February 1998     

Large Deviations of Heavy-Tailed Sums with Applications in Insurance

First we give a short review of large deviation results for sums of i.i.d. random variables. The main emphasis is on heavy-tailed distributions. We stress more the methodology than the detailed calculations. Large deviation techniques are then applied to randomly indexed sums and shot noise processes. We also indicate the close relationship between large deviation results and the modeling of large insurance claims. This revised version was published online in July 2006 with corrections to the Cover Date.     

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。     

Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions<Superscript>∗</Superscript>

We improve bounds of accuracy of the normal approximation to the distribution of a sum of independent random variables under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. We extend these results to Poisson binomial, binomial, and Poisson random sums. Under the same conditions, we obtain bounds for the accuracy of approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, we construct these bounds for the accuracy of approximation of the distributions of geometric, negative binomial, and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma, and Student distributions, respectively.     

L族END的多元部分和的精确大偏差下界

从保险的实际出发,研究服从长尾分布族(L族)上的多元风险模型中随机变量序列的部分和的精确大偏差,其中假设随机变量序列是一列延拓负相依(END)的、同分布的随机变量序列,利用基于求L族的精确大偏差的方法得到了随机变量部分和的渐近下界.     

Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables

We derive exponential bounds for the tail of the distribution of normalized sums of triangular arrays of random variables, not necessarily independent, under the law of ordinary logarithm.

Furthermore, we provide estimates for partial sums of triangular arrays of independent random variables belonging to suitable grand Lebesgue spaces and having heavy-tailed distributions.

     

Poissonian subordinators,the Wiener–Ornstein–Uhlenbeck field,and a relation between the Ornstein–Uhlenbeck processes and Brownian bridges

We apply to a sequence of i.i.d. random variables a time change operator via a Poisson process that is independent of this sequence. We consider sums of independent copies of processes constructed in this way and having continuous time. Finite limit distributions of these sums coincide with the finite limit distributions of the Wiener–Ornstein–Uhlenbeck field that is the tensor product of a Brownian motion and the Ornstein–Uhlenbeck process. The transition characteristics of the limit Ornstein–Uhlenbeck process are described by Brownian bridges that are builded into the Wiener–Ornstein–Uhlenbeck field. Bibliography: 4 titles.     

Stochastic compactness of distributions of sums of independent random variables with finite variances<Superscript>∗</Superscript>

We consider sums of independent random variables within the scheme of series. We focus on the case where every summand has a zero mean and finite variance and sums have unit variances. We obtain a criterion of stochastic compactness (defined by W. Feller) for sequences of distributions of such sums. The condition of uniform asymptotic negligibility of summands is not supposed.     

Limiting behavior of weighted sums with stable distributions

We present an integral test to determine the limiting behavior of weighted sums of independent, symmetric random variables with stable distributions, and deduce Chover-type laws of the iterated logarithm for them.     

On the Distribution of Random Samples from Finite Populations of Random Variables

This article deals with probability distributions of sums of simple random sample and Bernoulli sample when samples are selected from finite population of independent random variables. Random variables are quasi-lattice. Probability distributions from class ? and Poisson distribution are used for approximation. Analogue of Cornish-Fisher transformation is obtained in case of limit distributions from class ?.     

Ordered Random Variables

We discuss methods, which enable us to express distributions of ordered random variables (classical order statistics, induced order statistics, records and generalized order statistics) via distributions of sums of independent terms.     

On closure properties of heavy-tailed distributions for random sums*

We investigate some closure properties of heavy-tailed distributions for random sums. We first consider the random sums with independent or some certain dependent long-tailed increments. We also consider the real-valued increments with dominatedly varying-tailed distributions under extended negative dependence and obtain two results on necessary and sufficient conditions or sufficient conditions for the closure on random sums.     

Some limit theorems for independent random processes at random points in time and random elements defined by them

The paper deals with random variables which are the values of independent identically distributed stochastic processes at random points in time. We obtain conditions for the weak convergence of their sums, at almost all points in time, to the same infinitely divisible distribution and describe the limit distribution for these sums. Also we obtain an analog of the Donsker theorem and limit theorems for empirical processes for such random variables. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-16099). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

A free analogue of Hincin's characterization of infinite divisibility

Hincin characterized the class of infinitely divisible distributions on the line as the class of all distributional limits of sums of infinitesimal independent random variables. We show that an analogue of this characterization is true in the addition theory of free random variables introduced by Voiculescu.

     

Semi-stable distributions in free probability theory

Semi-stable distributions, in classical probability theory, are characterized as limiting distributions of subsequences of normalized partial sums of independent and identically distributed random variables. We establish the noncommutative counterpart of semi-stable distributions. We study the characterization of noncommutative semi-stability through free cumulant transform and develop the free semi-stability and domain of semi-stable attraction in free probability theory.     

NA序列线性形式的强稳定性

本文研究了NA变量线性形式的强稳定性.利用随机变量的截尾术及强大数定律,得到了一般情况下不同分布NA变量具有强稳定性的充分条件,推广了NA列的Jamison型加权和具有强稳定性的充分条件.     

The convergence of the random lines defined by observations of a stochastic process

We deal with independent random variables which are the values of a stochastic process taken at random points in time. So we have random variables depending upon a random parameter. We obtain the conditions providing the weak convergence of random lines defined by sums or maxima or bilinear forms of these random variables for almost all values of the parameter, to one and the same stochastic process. These limit stochastic processes are described. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

On the real accuracy of approximation in the central limit theorem

We consider various approximations in the central limit theorem for distributions of sums of independent random variables. We study how many summands in the normalized sums guarantee the precision 10⁻³ for these approximations. It turns out that for the same distribution but different approximations this number varies from hundreds of thousands to a few tens.     

不同分布NA序列加权和的完全收敛性

本文讨论了不同分布NA随机变量序列加权和的完全收敛性，获得了较[7]中的定理1及定理A更为一般的安全收敛性，并得到了完全收敛速度与矩条件之间的等价关系。