### Journals Information

**
Mathematics and Statistics Vol. 10(1), pp. 262 - 268 DOI: 10.13189/ms.2022.100125 Reprint (PDF) (379Kb) **

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

**Azam A. Imomov ^{1}^{,2}^{,*}, Zuhriddin A. Nazarov ^{2}**

^{1}Department of Mathematics, Faculty of Physics and Mathematics, Karshi State University, Karshi, Uzbekistan

^{2}V.I.Romanovskiy Institute of Mathematics, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan

**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.

**KEYWORDS**

Random Variables, Lindeberg Condition, Sum in a Special Form, Law of Large Numbers, Central Limit Theorem, Stochastic Branching Systems

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Azam A. Imomov , Zuhriddin A. Nazarov , "Limit Theorems for The Sums of Random Variables in A Special Form," Mathematics and Statistics, Vol. 10, No. 1, pp. 262 - 268, 2022. DOI: 10.13189/ms.2022.100125.

(b). APA Format:

Azam A. Imomov , Zuhriddin A. Nazarov (2022). Limit Theorems for The Sums of Random Variables in A Special Form. Mathematics and Statistics, 10(1), 262 - 268. DOI: 10.13189/ms.2022.100125.