Limiting behavior of random permanents
type: Item
instance of: scholarly article
title: Limiting behavior of random permanents
zbMATH Open document ID: 0948.60015
DOI: 10.1016/S0167-7152(99)00054-1
author: Jacek Wesołowski
author: Grzegorz A. Rempala
published in: Statistics \& Probability Letters
publication date: 2000-11-07T00:00:00Z
review text: Given a matrix \(X=[X_{i,j}]\), \(i=1,\dots, m\), \(j=1,\dots, n,\) where \(X_{i,j}\) are i.i.d. r.v.'s with \(EX_{1,1}=\mu\neq 0\) and \(\operatorname {var} X_{1,1}=\sigma^2\), let \(\operatorname {Per} (X)\) be the permanent of \(X\) and set \(S_{m,n}= \operatorname {Per} (X)[\nu m\mu!\mu^m]^{-1}\), \(\gamma=\sigma/\mu.\) If \(m,n\to\infty\) so that \(m/n\to\lambda >0\), then \(S_{m,n}@>D>> \exp(\sqrt{\lambda}\gamma N-\lambda\gamma^2/2),\) \(N\) being a standard normal r.v., and if \(\lambda =0\), then \(\gamma^{-1}\sqrt{n/m}(S_{m,n}-1) @>D>> N.\) The proofs are based on analysis of variance of random permanents and involve related asymptotic properties of elementary symmetric polynomials in \(X_{i,j}.\) The results of \textit{V. S. Korolyuk} and \textit{Yu. V. Borovskikh} [in: Probability theory and mathematical statistics, 176-187 (1992)] and \textit{A. J. van Es} and \textit{R. Helmers} [Probab. Theory Relat. Fields 80, No. 1, 21-35 (1988; Zbl 0638.60028)] are employed.
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1367949
zbMATH Keywords: random permanent
zbMATH Keywords: elementary symmetric polynomial
zbMATH Keywords: central limit theorem
reviewed by: Alexander V. Bulinski
MaRDI profile type: Publication
cites work: Random permanents and symmetric statistics
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description: scientific article; zbMATH DE number 1367949