On sums of independent random variables whose distributions belong to the max domain of attraction of max stable laws

In this article we study the max domains of attraction of distributions of sums of independent random variables belonging to the max domains of attraction of max stable laws under linear normalization. These results lead to the study of the max domains of attraction of distributions of products of independent random variables belonging to the max domain of attraction of max stable laws under power normalization.     

Large deviations of sums of independent random variables from the domain of attraction of a stable law

The paper continues the author's previous paper and deals with the case where the existence of the exponential moment of the distribution under consideration is not assumed. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 262–283.     

On the law of the iterated logarithm for increments of sums of independent random variables

We obtain the law of the iterated logarithm for increments of sums of independent random variables. Our results generalize the Kolmogorov theorem and the Hartman—Wintner theorem on the law of the iterated logarithm. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 174–186.     

An almost sure limit theorem for random sums of independent random variables in the domain of attraction of a semistable law

In this paper we prove an almost sure limit theorem for random sums of independent random variables in the domain of attraction of a p-semistable law and describe the limit law.     

A limit theorem for the moments of sums of independent random variables

Forn≧1, letS _n=ΣX _n,i (1≦i≦r _n <∞), where the summands ofS _n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some for allt≧1 and all values ofx. Theorem.For centering constants c _n,let S _n − c _n converge in distribution to a random variable S. (A)In order that Ef(S_n − c_n) converge to a limit L, it is necessary and sufficient that there exist a common limit (B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.     

On the growth of sums of independent random variables

Some estimates of the growth of sums of independent random variables almost surely are established without any moment conditions. Bibliography: 6 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 158–164.This research was partially supported by the Russian Foundation for Basic Research, grant 02-01-00779, and by the Program Leading Scientific Schools, grant 00-15-96019.Translated by V. V. Petrov.     

离散随机序列加权和的若干极限定理

设{Xn,n≥0}是任意离散随机变量序列,{ank,0≤k≤n,n≥0)是一常数阵列,我们引入随机序列渐近对数似然比的概念,作为表征随机序列的真实概率测度P与参考测度Q之间的差异的度量,用分析方法,得到了随机序列Jamison型加权和的若干随机偏差定理.     

On the rate of convergence in the global central limit theorem for random sums of independent random variables

We present upper bounds of the integral \( {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathbf{P}\left\{{Z}_N<x\right\}-\varPhi (x)\right|\mathrm{d}x \) for 0 ≤ l ≤ 1 + δ, where 0 < δ ≤ 1, Φ(x) is a standard normal distribution function, and Z _N = \( {S}_N/\sqrt{\mathbf{V}{S}_N} \) is the normalized random sum with variance V S _N > 0 (S _N = X ₁ + · · · + X _N ) of centered independent random variables X ₁ ,X ₂ , . . . . The number of summands N is a nonnegative integer-valued random variable independent of X ₁ ,X ₂ , . . . .     

Limit behavior of sums of a random number of independent random variables

Lithuanian Mathematical Journal -     

Limit behavior of centered random sums of independent identically distributed random variables

Under different assumptions, necessary and sufficient conditions are given for the weak convergence of sums of a random number of independent identically distributed random variables. Supported by the Russian Foundation for Fundamental Researches (grant No. 93-011-1446). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.