Ratio ergodicity for non-homogeneous Markov chains in general state spaces
type: Item
instance of: scholarly article
title: Ratio ergodicity for non-homogeneous Markov chains in general state spaces
zbMATH Open document ID: 0813.60059
DOI: 10.1007/BF02213452
author: Isidore Fleischer
author: Anatole Joffe
published in: Journal of Theoretical Probability
publication date: 1995-02-26T00:00:00Z
review text: Suppose \(T\) is an infinite nonnegative matrix satisfying \(T \text{\textbf{1}} = \text{\textbf{1}}\) (that is, a stochastic matrix) with no zero columns. Then it was shown by the reviewer [ibid. 6, No. 2, 345-352 (1993; Zbl 0774.60013)] that \[ \tau_ 1 (T) \leq \overline \tau_ B (T') \tag \(*\){} \] where \(\tau_ 1\) is Dobrushin's coefficient of ergodicity, and \(\overline \tau_ B\) is Birkhoff's as defined by \textit{S. Gilbert} and \textit{A. Mukherjea} [Math. Z. 196, 485-490 (1987; Zbl 0623.15013)] for a nonnegative matrix \(A\) whose columns are uniformly bounded from zero and infinity, and which has no zero rows, by taking \(A=T'\). The authors extend \((*)\) to a general positive operator \(T\) on a space of measures, where \(T\) preserves the constant function \textbf{1} and has positive adjoint \(T'\), and \(\overline \tau_ B\) is defined, more generally, so that \(\overline \tau_ B(T) = \overline \tau_ B(T')\). A corresponding extension to general state spaces of the application to inhomogeneous Markov chains is made.
reviewed by: Eugene Seneta
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 727273
zbMATH Keywords: Dobrushin's coefficient of ergodicity
zbMATH Keywords: positive operator
zbMATH Keywords: space of measures
zbMATH Keywords: Markov chains
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02213452
OpenAlex ID: W1976234294
description: scientific article; zbMATH DE number 727273