Limit Theorems for The Sums of Random Variables in A Special Form

Zuhriddin A Nazarov,Azam A Imomov

doi:10.13189/ms.2022.100125

Abstract

In this paper, we consider some functionals of the sums of independent identically distributed random variables. The functionals of the sums are important in probabilistic models and stochastic branching systems. In connection with the application in various probabilistic models and stochastic branching systems, we are interested in the fulfillment of the law of large numbers and the Central limit theorem for these sums. The main hypotheses of the paper are the presence of second order moments of the variables and the fulfillment of the Lindeberg condition is considered. The research object and subject of this paper consists of specially generated random variables using the sums of non-bound random variables. In total, 6 different sums in a special form were studied in the paper and this sum was not previously studied by other scientists. The purpose of the paper is to examine whether these sums in a special form satisfy the terms of the law of large numbers and the Central limit theorem. The main result of the paper is to show that the law of large numbers and the terms of the classical limit theorem are fulfilled in some cases. The results obtained in the paper are of theoretical importance, The Central limit theorem analogues proved here are applications of Lindeberg theorem. The results can be applied to the determination of the fluctuation of immigration branching systems as well as the asymptotic state of autoregression processes. At the same time, from the main results obtained in the paper it can be used in practical lessons conducted on the theory of probability. The results of the paper will be an important guide for young researchers. Important theorems proved in the paper can be used in probability theory, stochastic branching systems and other practical problems.

Highlights

Let N be set of natural numbers and {ξn, n ∈ N} – a sequence of random variables
In connection with the application in various probabilistic models and stochastic branching systems, we are interested in the fulfillment of the law of large numbers and the Central limit theorem for these sums
The purpose of the paper is to examine whether these sums in a special form satisfy the terms of the law of large numbers and the Central limit theorem

Summary

Introduction

Let N be set of natural numbers and {ξn, n ∈ N} – a sequence of random variables. The solution of important problems in many areas of probability theory and mathematical statistics leads to the determination of the asymptotic state of the sum. Xn − EXn = mn−kMk, k=1 where m is the per mean capita number of ”aborigines” and Mk is martingale difference; see for instance [4] In this model, it will be necessary to study the asymptotes of the random variables n k=1 fnk (Mk ). The study of limit theorems for a sum of random variables is one of the main problems of probability theory and mathematical statistics In this field, significant results have been achieved by P.L.Chebyshev, A.N.Kolmogorov, B.V.Gnedenko, A.Ya.Khinchin, W.Feller, A.V.Prokhorov, Ya.V.Lindeberg, V.M.Zolotarev and etc.; see [5], [6]. The main results obtained are of theoretical importance and can be applied to determine fluctuations of stochastic branching-immigration systems, as well as the asymptotic behavior of autoregressive processes

The law of large numbers and the strong law of large numbers

Central limit theorem

Sums of random variables in a special form and asymptotes of their variances

Central limit theorem analogues

Findings

Proof of Theorems