A transient random walk on stochastic matrices with Dirichlet distributions
type: Item
instance of: scholarly article
title: A transient random walk on stochastic matrices with Dirichlet distributions
zbMATH Open document ID: 0807.60066
author: Jean-François Chamayou
author: Gérard Letac
published in: The Annals of Probability
publication date: 1994-06-16T00:00:00Z
review text: Suppose \(X_ 1, X_ 2, \ldots, X_ n, \ldots\) are stochastic \(d \times d\) matrices, which are random in the sense that \(X_ k\) has \(d\) independent rows and each row has a Dirichlet distribution. If \(X_ n\) \((n \geq 1\)) are independent and identically distributed, then it is shown that the product \(X_ n X_{n - 1} \cdots X_ 1\) converges almost surely to a random stochastic matrix with \(d\) identical rows which also have a Dirichlet distribution. There is a symmetry condition: if the \(d\) parameters of the Dirichlet distribution of the rows of \(X_ k\) are put in a \(d \times d\) matrix \(A\), then it is assumed that the sum of the entries on row \(j\) is the same as the sum of the entries on column \(j\). The parameters of the limiting Dirichlet distribution are then given by these \(d\) row (or column) sums. This generalizes a result of the reviewer [J. Appl. Probab. 23, 1019-1024 (1986; Zbl 0606.60012)] who treated the case \(d = 2\) for a matrix \(A\) with four identical entries.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60E10
zbMATH DE Number: 567191
zbMATH Keywords: Dirichlet distribution
zbMATH Keywords: products of random matrices
reviewed by: Walter Van Assche
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988865
OpenAlex ID: W2002546119
DOI: 10.1214/AOP/1176988865
description: scientific article; zbMATH DE number 567191