Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 3, pp. 588-592
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A. V. Pečinkin. Limit distribution for a random walk with absorption. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 3, pp. 588-592. http://geodesic.mathdoc.fr/item/TVP_1980_25_3_a11/
@article{TVP_1980_25_3_a11,
author = {A. V. Pe\v{c}inkin},
title = {Limit distribution for a~random walk with absorption},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {588--592},
year = {1980},
volume = {25},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_3_a11/}
}
TY - JOUR
AU - A. V. Pečinkin
TI - Limit distribution for a random walk with absorption
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1980
SP - 588
EP - 592
VL - 25
IS - 3
UR - http://geodesic.mathdoc.fr/item/TVP_1980_25_3_a11/
LA - ru
ID - TVP_1980_25_3_a11
ER -
%0 Journal Article
%A A. V. Pečinkin
%T Limit distribution for a random walk with absorption
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1980
%P 588-592
%V 25
%N 3
%U http://geodesic.mathdoc.fr/item/TVP_1980_25_3_a11/
%G ru
%F TVP_1980_25_3_a11
Let $\xi_1,\xi_2,\dots$ are independent identically distributed random variables, $\mathbf M\xi_1=0$, $\mathbf D\xi_1=1$ and $$ S_n=n^{-1/2}(\xi_1+\dots+\xi_n),\qquad\nu=\min\{n:S_n<0\}. $$ We show that $$ \mathbf P\{S_\nu<x\mid\nu>n\}\to V(x),\qquad\mathbf P\{S_n<x\mid\nu>n\}\to 1-e^{-x^2/2}. $$
