The law of large numbers for permanents of random matrices

A. N. Timashev

Diskretnaya Matematika, Tome 17 (2005) no. 4, pp. 59-71

Résumé

We consider the class of random matrices $C=(c_{ij})$, $i,j=1,\dots,N$, whose elements are independent random variables distributed by the same law as a certain random variable $\xi$ such that $\mathsf E\xi^2>0$. As usual, $\operatorname{per}C$ stands for the permanent of the matrix $C$. In the triangular array series where $\xi=\xi_N$, $\mathsf E\xi_N\neq 0$, $N=1,2,\dotsc$, $\mathsf D\xi_N=o((\mathsf E\xi_N)^2)$ as $N\to\infty$, we prove that the sequence of random variables $\operatorname{per}C/(N!\,(\mathsf E\xi_N)^N)$ converges in probability to one as $N\to \infty$. 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 $\mu$ 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 $\operatorname{per}C/\mathsf E\operatorname{per}C$ in the cases where the vector $\mu$ coincides with the vector of frequencies of outcomes of the equiprobable polynomial scheme with $N$ outcomes and $n$ trials and also where $\mu$ is a random equiprobable solution of the equation $k_1+\ldots+k_N=n$ in non-negative integers $k_1,\dots,k_N$.