Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 1, pp. 18-26
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A. I. Martikaǐnen; V. V. Petrov. On necessary and sufficient conditions for the law of the iterated logarithm. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 1, pp. 18-26. http://geodesic.mathdoc.fr/item/TVP_1977_22_1_a1/
@article{TVP_1977_22_1_a1,
author = {A. I. Martikaǐnen and V. V. Petrov},
title = {On necessary and sufficient conditions for the law of the iterated logarithm},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {18--26},
year = {1977},
volume = {22},
number = {1},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_1_a1/}
}
TY - JOUR
AU - A. I. Martikaǐnen
AU - V. V. Petrov
TI - On necessary and sufficient conditions for the law of the iterated logarithm
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1977
SP - 18
EP - 26
VL - 22
IS - 1
UR - http://geodesic.mathdoc.fr/item/TVP_1977_22_1_a1/
LA - ru
ID - TVP_1977_22_1_a1
ER -
%0 Journal Article
%A A. I. Martikaǐnen
%A V. V. Petrov
%T On necessary and sufficient conditions for the law of the iterated logarithm
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1977
%P 18-26
%V 22
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_1977_22_1_a1/
%G ru
%F TVP_1977_22_1_a1
Let $\{X_n;\,n=1,2,\dots\}$ be a sequence of independent not necessarily identically distributed random variables and $\{a_n;\,n=1,2,\dots\}$ be a non-decreasing sequence of positive numbers such that $a_n\to\infty$. We put $\displaystyle S_n=\sum_{j=1}^n X_j$. Necessary and sufficient conditions are found for the relations $\limsup(S_n/a_n)\le 1$ a.s. and $\limsup(S_n/a_n)=1$ a.s. No assumptions about existence of any moments are made.
