Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 1, pp. 156-160
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A. G. Postnikov; A. A. Judin. On an estimate of the concentration function for the sum of identically distributed two-dimensional independent lattice random vectors. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 1, pp. 156-160. http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a13/
@article{TVP_1981_26_1_a13,
author = {A. G. Postnikov and A. A. Judin},
title = {On an estimate of the concentration function for the sum of identically distributed two-dimensional independent lattice random vectors},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {156--160},
year = {1981},
volume = {26},
number = {1},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a13/}
}
TY - JOUR
AU - A. G. Postnikov
AU - A. A. Judin
TI - On an estimate of the concentration function for the sum of identically distributed two-dimensional independent lattice random vectors
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1981
SP - 156
EP - 160
VL - 26
IS - 1
UR - http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a13/
LA - ru
ID - TVP_1981_26_1_a13
ER -
%0 Journal Article
%A A. G. Postnikov
%A A. A. Judin
%T On an estimate of the concentration function for the sum of identically distributed two-dimensional independent lattice random vectors
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1981
%P 156-160
%V 26
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a13/
%G ru
%F TVP_1981_26_1_a13
The following theorem is proved. If $\xi_1,\xi_2,\dots$ is a sequence of non-degenerate identically distributed independent random variables with values in $Z^2$, then $$ \sup_{m\in Z^2}\mathbf P(\xi_1+\dots+\xi_n=m)\le Cn^{-1}\Delta^{-1/2}, $$ where $C$ is an absolute constant, $\Delta=(P_L-P_0)(1-P_L)$, $$ P_0=\max_{m\in Z^2}\mathbf P\{\xi=x\},\qquad P_L=\max_H\mathbf P\{\xi\in H\}, $$$H$ is a set of points belonging to some straight line.
