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.     

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.     

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.     

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.     

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.     

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

Lithuanian Mathematical Journal -     

Remarks on summability of series formed of deviation probabilities of sums of independent identically distributed random variables

We make some remarks leading to a refinement of the recent work of Klesov (1993) on the connection between the convergence of the series $\Sigma _{n = 1}^\infty \tau _n P(|S_n | \ge \varepsilon n^\alpha )$ for every ε > 0 and the convergence of $\Sigma _{n = 1}^\infty n\tau _n P(|X_1 | \ge \varepsilon n^\alpha )$ again for every ε > 0.     

A general Hsu-Robbins-Erdős Type estimate of tail probabilities of sums of independent identically distributed random variables

Periodica Mathematica Hungarica - Let X 1,X 2,... be a sequence of independent and identically distributed random variables, and put % MATHTYPE!MTEF!2!1!+-%...     

Estimation of proximity of distributions of sequential sums of independent identically distributed random vectors

Let be a distribution in, let be the n-fold convolution of with itself, and let. It is proved that where depends on certain characteristics of.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 83–87, 1980.In conclusion, the author expresses his gratitude to I. A. Ibragimov for his interest in the paper.     

Asymptotic behavior for sums of non-identically distributed random variables

For any given positive integer m, let X_i, 1 ≤ i ≤ m be m independent random variables with distributions F_i, 1 ≤ i ≤ m. When all the summands are nonnegative and at least one of them is heavy-tailed, we prove that the lower limit of the ratio ■equals 1 as x →∞. When the summands are real-valued, we also obtain some asymptotic results for the tail probability of the sums. Besides, a local version as well as a density version of the above results is also presented.