Existence conditions of permanental and multivariate negative binomial distributions
type: Item
instance of: scholarly article
title: Existence conditions of permanental and multivariate negative binomial distributions
zbMATH Open document ID: 1385.60049
DOI: 10.1214/17-AOP1179
author: Nathalie Eisenbaum
author: Franck Maunoury
published in: The Annals of Probability
publication date: 2018-02-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.aop/1513069273
review text: We say that a matrix \(A\) is \(\beta\)-positive definite if the multivariate Taylor expansion in \(z_1^{n_1}\dots z_d^{n_d}\) of \(\det(I-Z A)^{-\beta}\), where \(Z\) is a diagonal matrix with elements \(z_1\), \(\dots\), \(z_d\), has only nonnegative coefficients. In this manuscript, the authors correct some older results about the \(\beta\)-positive definite matrix and \(\beta\)-permanental matrix and introduce some new results. They first correct the criteria introduced in [\textit{R. C. Griffiths} and \textit{R. K. Milne}, J. Multivariate Anal. 22, 13--23 (1987; Zbl 0618.62060)] and use it to obtain some results about \(\beta\)-positivity. The rest of the manuscript is dedicated on \(\beta\)-permanental matrix of dimensions below and above \(3\).
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60E10
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 6838133
zbMATH Keywords: permanental vector
zbMATH Keywords: negative binomial distribution
zbMATH Keywords: \(M\)-matrix
zbMATH Keywords: Gaussian vector
zbMATH Keywords: matrix cycles
zbMATH Keywords: permanent
zbMATH Keywords: determinant
zbMATH Keywords: infinite divisibility
reviewed by: Miroslav M. Ristić
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6838133