Limit theorems for random processes constructed from sums of independent identically distributed random variables

There are proved limit theorems for random processes constructed from sums of independent identically distributed random variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 141–145, January, 1991.     

Approximation for Abel sums of independent,identically distributed random variables

This note represents a probability inequality for an approximation of Abel sums of independent, identically distributed random variables. This approximation implies a rate for the Prohorov-Lévy distance between Abel sums and their limit process.     

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.     

Limit theorems of general functions of independent and identically distributed random variables

In this paper we derive limit theorems of some general functions of independent and identically distributed random variables. A stability property is used to derive the limit theory for general functions. A procedure followed in de Haan (1976) and Leadbetter et al. (1983) is used to prove the main result. The limit theorems for the maximum, minimum and sum of fixed sample sizes and random sample sizes are derived as special cases of the main result.     

On the invariance principle for sums of independent identically distributed random variables

The paper deals with the invariance principle for sums of independent identically distributed random variables. First it compares the different possibilities of posing the problem. The sharpest results of this theory are presented with a sketch of their proofs. At the end of the paper some unsolved problems are given.     

独立情形下自正则和的精确渐近性

研究均值为零非退化的独立同分布的随机变量序列正则和收敛性,在适当条件下,获得了自正则和精确渐近性的一般结果.     

A limit theorem for multiple sums of identically distributed independent random variables

One investigates the asymptotic behavior (with respect to a small parameter) of series of probabilities of large deviations for sums of random variables with multidimensional indices.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 265–277, 1986.     

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.     

Bootstraps of sums of independent but not identically distributed stochastic processes

A central limit theorem is developed for sums of independent but not identically distributed stochastic processes multiplied by independent real random variables with mean zero. Weak convergence of the Hoffmann–Jørgensen–Dudley type, as described in van der Vaart and Wellner (Weak Convergence and Empirical Processes, Springer, New York, 1996), is utilized. These results allow Monte Carlo estimation of limiting probability measures obtained from application of Pollard's (Empirical Processes: Theory and Applications, IMS, Hayward, CA, 1990) functional central limit theorem for empirical processes. An application of this theory to the two-parameter Cox score process with staggered entry data is given for illustration. For this process, the proposed multiplier bootstrap appears to be the first successful method for estimating the associated limiting distribution. The results of this paper compliment previous bootstrap and multiplier central limit theorems for independent and identically distributed empirical processes.     

The limiting behaviour of the last exit time for sequences of independent,identically distributed random variables

Summary Let {X _k , k0} be i.i.d. random variables with EX ⁺< and define t =max{k0: X _k > k} if such a k exists and =0 else, the last exit time of the sequence X _k for fixed >0. We discuss weak limit laws for t as 0; in particular the limit distributions, the stability and the relative stability.This work is partially supported by a grant of the Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung, while the author was at the University of Pittsburgh, USAHerrn Prof. L. Schmetterer zu seinem 60. Geburtstag gewidmet     

On the average number of records for sequences of not identically distributed random variables

The scheme of n series of independent random variables X ₁₁, X ₂₁, …, X _k1, X ₁₂, X ₂₂, …, X _k2, …, X _1n , X _2n , …, X _kn is considered. Each of these successive series X _1m , X _2m , …, X _km , m = 1, 2, …, n consists of k variables with continuous distribution functions F ₁, F ₂, …, F _k , which are the same for all series. Let N(nk) be the number of upper records of the given nk random variables, and EN(nk) be the corresponding expected value. For EN(nk) exact upper and lower estimates are obtained. Examples are given of the sets of distribution functions for which these estimates are attained.