Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 3, pp. 497-500
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B. A. Rogozin. The accuracy of approximation oi the limit distribution to the distribution of the maximum of sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 3, pp. 497-500. http://geodesic.mathdoc.fr/item/TVP_1966_11_3_a8/
@article{TVP_1966_11_3_a8,
author = {B. A. Rogozin},
title = {The accuracy of approximation oi the limit distribution to the distribution of the maximum of sums of independent random variables},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {497--500},
year = {1966},
volume = {11},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1966_11_3_a8/}
}
TY - JOUR
AU - B. A. Rogozin
TI - The accuracy of approximation oi the limit distribution to the distribution of the maximum of sums of independent random variables
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1966
SP - 497
EP - 500
VL - 11
IS - 3
UR - http://geodesic.mathdoc.fr/item/TVP_1966_11_3_a8/
LA - ru
ID - TVP_1966_11_3_a8
ER -
%0 Journal Article
%A B. A. Rogozin
%T The accuracy of approximation oi the limit distribution to the distribution of the maximum of sums of independent random variables
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1966
%P 497-500
%V 11
%N 3
%U http://geodesic.mathdoc.fr/item/TVP_1966_11_3_a8/
%G ru
%F TVP_1966_11_3_a8
Let $\xi_1\xi_2,\dots$ be a sequence of identically distributed independent random variables n and $S_0=0$, $S_n=\sum_{k=1}^n\xi_k$, $n=1,2,\dots$, $\bar S_n=\max_{0\le k\le n}S_k$, $n=0,1\dots$. Let us suppose that $\mathbf M\xi_1=a>0$, $\beta_3=\mathbf M|\xi_1-a|^3<\infty$, and denote $\sigma^2=\mathbf M(\xi_1-a)^2$. It is established that $$ \mathbf P\{S_n\le x\}-\mathbf P\{\bar S_n\le x\}\le\frac C{\sqrt n}\max\biggl\{\frac{\beta_3^2}{\sigma^6},\frac{\beta_3^2}{a^6},\frac{(\mathbf M|\xi_1|)^2}{\sigma^2}\biggr\} $$ where $С$ is a constant.
