A theorem on the distribution of the rank of a sparse Boolean random matrix and some applications

Abstract

Abstract. We consider some estimates of the rate of convergence of the distribution of a sparse Boolean random matrix to the Poisson distribution. The results obtained in the paper are applied to estimate the probability that a nonhomogeneous system of Boolean random linear equations is consistent. IntroductionDifferent methods are used in [1] and [2] to prove that the distribution of the rank r(A) of a sparse Boolean random matrix A approaches the Poisson distribution as T = T (n) → ∞ and n → ∞ (here n and T mean the number of columns and rows in the matrix A, respectively). If T and n are finite, then the distribution of the rank r(A) can be expressed in terms of the factorial moments of the random variable r(A). These expressions are found in [3]. The asymptotic behavior as n → ∞ of the factorial moments is given in [4, Theorem 4] under certain assumptions, say if n − T = const and if the distribution of entries of the matrix A depends on the number of the column only.We are interested in the bounds of the rate of convergence of the distribution of the rank of the matrix A to the Poisson distribution as n grows and in applications of these bounds for an estimation of the probability that the following nonhomogeneous system of linear equations is consistent:where the entries of the matrix A of coefficients belong to the field GF (2) consisting of two elements, the vector column B = (b 1 , . . . , b T ) is formed by the random variables b 1 , . . . , b T that do not depend on the entries of the matrix A and moreover b 1 , . . . , b T are jointly independent random variables assuming values in GF (2) and having known distributions, and where X = (x 1 , x 2 , . . . , x n ) is an n-dimensional column vector such that x i ∈ GF (2), i = 1, . . . , n.In contrast to the papers [1, 2], we consider random matrices A whose entries have the distributions that may depend on their position in the matrix A.2000 Mathematics Subject Classification. Primary 68U20; Secondary 60G10. Key words and phrases. Boolean random matrix, rank of a matrix, the probability that a system is consistent, the rate of convergence of distributions.