Geodesic
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
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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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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/

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