Convergence parameter associated with a Markov chain and a family of functions
type: Item
instance of: scholarly article
title: Convergence parameter associated with a Markov chain and a family of functions
zbMATH Open document ID: 1205.60136
DOI: 10.1134/S0001434610010347
published in: Mathematical Notes
publication date: 2010-09-24T00:00:00Z
review text: The definition proposed of a convergence parameter \(R\left( W\right) \) associated with a Markov chain \(X\) with a measurable state space \(\left( E,\mathcal{B}\right) \) and any nonempty set \(W \) of bounded below measurable functions \(f:E\rightarrow \mathbb{R} \), is wider than the well-known definition of a convergence parameter \(R\) in the sense of D. Vere-Jones, R. L. Tweedie or E. Nummelin. Very often, \(R\left( W\right) <\infty \), and there exists a set playing the role of the absorbing set in Nummelin's definition of \(R\). Special attention is paid to the case in which \(E\) is locally compact, \(X\) is a Feller chain on \(E\), and \(W\) coincides with the family \(\varepsilon _{0}^{+}\) of all compactly supported continuous functions \(f\geq 0\left( f\equiv \!\!\!\!\!/0\right) .\) In particular, certain conditions for \(R\left( \varepsilon _{0}^{+}\right) ^{-1} \) to coincide with the norm of an appropriate modification of the chain transition operator are found.
reviewed by: Marius Iosifescu
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 5791046
zbMATH Keywords: convergence parameter
zbMATH Keywords: Markov chain
zbMATH Keywords: absorbing set
zbMATH Keywords: locally compact set
zbMATH Keywords: random walk
zbMATH Keywords: irreducible chain
zbMATH Keywords: Feller chain
zbMATH Keywords: measurable state space
author: M. G. Shur
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5791046