Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 369-372
Citer cet article
S. V. Nagaev. On the asymptotic behaviour of one-sided large deviation probabilities. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 369-372. http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a9/
@article{TVP_1981_26_2_a9,
author = {S. V. Nagaev},
title = {On the asymptotic behaviour of one-sided large deviation probabilities},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {369--372},
year = {1981},
volume = {26},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a9/}
}
TY - JOUR
AU - S. V. Nagaev
TI - On the asymptotic behaviour of one-sided large deviation probabilities
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1981
SP - 369
EP - 372
VL - 26
IS - 2
UR - http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a9/
LA - ru
ID - TVP_1981_26_2_a9
ER -
%0 Journal Article
%A S. V. Nagaev
%T On the asymptotic behaviour of one-sided large deviation probabilities
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1981
%P 369-372
%V 26
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a9/
%G ru
%F TVP_1981_26_2_a9
Let $X_1,X_2,\dots$ be i. i. d. random variables, $$ \mathbf EX_1=0,\qquad F(x)=\mathbf P\{X_1<x\},\qquad S_n=X_1+\dots+X_n $$ We prove that if for $x\to\infty$$$ 1-F(x)\thicksim x^{-\alpha}h(x),\qquad \alpha>1, $$ where $h(x)$ is a slowly varying function, then $$ \mathbf P\{S_n\ge x\}\thicksim n(1-F(x))\qquad\text{for}\ n\to\infty\ \text{and}\ \liminf_{n\to\infty}x/n>0. $$
