Diskretnaya Matematika, Tome 19 (2007) no. 1, pp. 17-26
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V. G. Mikhailov. Limit theorems for the number of solutions of a system of random linear equations belonging to a given set. Diskretnaya Matematika, Tome 19 (2007) no. 1, pp. 17-26. http://geodesic.mathdoc.fr/item/DM_2007_19_1_a3/
@article{DM_2007_19_1_a3,
author = {V. G. Mikhailov},
title = {Limit theorems for the number of solutions of a~system of random linear equations belonging to a~given set},
journal = {Diskretnaya Matematika},
pages = {17--26},
year = {2007},
volume = {19},
number = {1},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/DM_2007_19_1_a3/}
}
TY - JOUR
AU - V. G. Mikhailov
TI - Limit theorems for the number of solutions of a system of random linear equations belonging to a given set
JO - Diskretnaya Matematika
PY - 2007
SP - 17
EP - 26
VL - 19
IS - 1
UR - http://geodesic.mathdoc.fr/item/DM_2007_19_1_a3/
LA - ru
ID - DM_2007_19_1_a3
ER -
%0 Journal Article
%A V. G. Mikhailov
%T Limit theorems for the number of solutions of a system of random linear equations belonging to a given set
%J Diskretnaya Matematika
%D 2007
%P 17-26
%V 19
%N 1
%U http://geodesic.mathdoc.fr/item/DM_2007_19_1_a3/
%G ru
%F DM_2007_19_1_a3
We investigate the asymptotic behaviour of the distribution of the number $\xi(B)$ of the solutions of a system of homogeneous random linear equations $Ax=0$ (the $T\times n$ matrix $A$ is composed of independent random variables $a_{i,j}$ uniformly distributed on a set of elements of a finite field $K$) which belong to some given set $B$ of non-zero $n$-dimensional vectors over the field $K$. We consider the case where, under a concordant growth of the parameters $n,T\to\infty$ and variations of the sets $B_1,\dots,B_s$ such that the mean values converge to finite limits, the limit distribution of the vector $(\xi(B_1),\dots,\xi(B_s))$ is an $s$-dimensional compound Poisson distribution. We give sufficient conditions for this convergence and find parameters of the limit distribution. We consider in detail the special case where $B_k$ is the set of vectors which do not contain a certain element $k\in K$.
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