The law of large numbers for permanents of random matrices

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We consider the class of random matrices C = (cij), i, j = 1,…N, whose elements are independent random variables distributed by the same law as a certain random variable ξ such that Eξ2 > 0. As usual, per C stands for the permanent of the matrix C. In the triangular array series where ξ = ξN, EξN ≠ 0, N = 1, 2, . . . , DξN = o((Eξ N)2)as N → ∞, we prove that the sequence of random variables per C/(N!(EξN)N) converges in probability to one as N → ∞. A similar result is shown to be true in a more general case where the rows of the matrix C are independent N-dimensional random vectors which have the same distribution coinciding with the distribution of a random vector μ whose components are identically distributed but are, generally speaking, dependent. We give sufficient conditions for the law of large numbers to be true for the sequence per C/E per C in the cases where the vector μ coincides with the vector of frequencies of outcomes of the equiprobable polynomial scheme with N outcomes and n trials and also where μ is a random equiprobable solution of the equation k1 + … + kN = n in non-negative integers k1,…,kN.