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 Diskr. Mat., 2005, Volume 17, Issue 4, Pages 59–71 (Mi dm129)

The law of large numbers for permanents of random matrices

A. N. Timashev

Abstract: We consider the class of random matrices $C=(c_{ij})$, $i,j=1,…,N$, 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 $k_1,…,k_N$.

DOI: https://doi.org/10.4213/dm129

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English version:
Discrete Mathematics and Applications, 2005, 15:5, 513–526

Bibliographic databases:

UDC: 519.2

Citation: A. N. Timashev, “The law of large numbers for permanents of random matrices”, Diskr. Mat., 17:4 (2005), 59–71; Discrete Math. Appl., 15:5 (2005), 513–526

Citation in format AMSBIB
\Bibitem{Tim05} \by A.~N.~Timashev \paper The law of large numbers for permanents of random matrices \jour Diskr. Mat. \yr 2005 \vol 17 \issue 4 \pages 59--71 \mathnet{http://mi.mathnet.ru/dm129} \crossref{https://doi.org/10.4213/dm129} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2240541} \zmath{https://zbmath.org/?q=an:05062017} \elib{https://elibrary.ru/item.asp?id=9154202} \transl \jour Discrete Math. Appl. \yr 2005 \vol 15 \issue 5 \pages 513--526 \crossref{https://doi.org/10.1515/156939205776368922}