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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

We investigate the asymptotic behaviour of the distribution of the number ξ(B) of the solutions of a system of homogeneous random linear equations Ax = 0 (the T × 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 → ∞ and variations of the sets B₁, . . . ,B_s such that the mean values converge to finite limits, the limit distribution of the vector (ξ(B₁), ... , ξ(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 ∈ K.

Discrete Mathematics and Applications, Walter de Gruyter

Print ISSN: 0924-9266
Volume: 17, 04/2007
Pages: 13 - 22

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