Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 4, pp. 710-718
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V. P. Čistyakov. Theorem on Sums of Independent Random Positive Variables and its Applications to Branching Processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 4, pp. 710-718. http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a12/
@article{TVP_1964_9_4_a12,
author = {V. P. \v{C}istyakov},
title = {Theorem on {Sums} of {Independent} {Random} {Positive} {Variables} and its {Applications} to {Branching} {Processes}},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {710--718},
year = {1964},
volume = {9},
number = {4},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a12/}
}
TY - JOUR
AU - V. P. Čistyakov
TI - Theorem on Sums of Independent Random Positive Variables and its Applications to Branching Processes
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1964
SP - 710
EP - 718
VL - 9
IS - 4
UR - http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a12/
LA - ru
ID - TVP_1964_9_4_a12
ER -
%0 Journal Article
%A V. P. Čistyakov
%T Theorem on Sums of Independent Random Positive Variables and its Applications to Branching Processes
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1964
%P 710-718
%V 9
%N 4
%U http://geodesic.mathdoc.fr/item/TVP_1964_9_4_a12/
%G ru
%F TVP_1964_9_4_a12
Let $\xi_1,\dots,\xi_n,\dots$ be independent random positive variables and let ${\mathbf P}\{\xi_k, $k=1,\dots,n,\dots$ Let us denote $$ {\mathbf P}\{\xi_1+\dots+\xi_n<t\}=G_n(t). $$ Theorem. $$ \lim_{t\to\infty}\frac{1-G_n(t)}{1-G(t)}=n,\qquad n=1,2,3,\dots, $$ and only if $$ \lim_{t\to\infty}\frac{1-G_2(t)}{1-G(t)}=2. $$ This theorem is useful in some investigations of age-dependent branching processes.
