ABOUT THE MAXIMUM SUM OF CASUAL NUMBER OF INDEPENDANT CASUAL QUANTITY

Abstract

Let ${\xi_n}$ be a sequence of independent random variables $S_n=\xi_1+\dots+\xi_n, ~\bar{S_n}=\max\limits_{1\leq k\leq n}|S_k|, ~n\geq1.$ It is shown that the correct variation of the function $N_n(t)=\sum\limits_{k=1}^n P(|\xi_k|\geq t)$ at infinity, where $P$ is the probability sign, implies the limit relations $\lim\limits_{t\rightarrow+\infty}\frac{P(\bar{S_n}\geq t)}{N_n(t)}= \lim\limits_{t\rightarrow+\infty}\frac{P(|S_n|\geq t)}{N_n(t)}=1.$ Let $v>0$ be a function independent of ${\xi_n}$ is an integer random variable with finite mathematical expectation, $$ S_v=\xi_1+\dots+\xi_v, ~\bar{S_v}=\max\limits_{1\leq n\leq v}|S_n|.$$ Consider the following model. Let ${\delta_n}$ be a sequence of positive numbers, $\alfa\geq0, ~L(t) $ vary slowly at infinity, $\lim\limits_{t\rightarrow+\infty}\big(P(|\xi_n|\geq t) \ t^{-\alfa}L(t)\big)= \delta_n$ uniformly on $n\geq1.$ Denote $c_n=(v=n), A_n=\sum\limits_{k=1}^n \delta_n,~ n\geq1,~A=\sum\limits_{n\geq 1}c_n \cdot A_n$. It has been proven that within the framework of the model $\lim\limits_{t\rightarrow +\infty}\frac{P(\bar{S_v}\geq t)}{ t^{-\alfa}L(t)}= \lim\limits_{t\rightarrow+\infty}\frac{P(|S_v|\geq t)}{ t^{-\alfa}L(t)}=A.$ Under the above conditions with existence of asymmetry in $P(\xi_n <x),~ n\geq1,$, asymmetry in $P(S_v<x)$ was found.