Tightness and continuity of a family of invariant measures for Markov chains depending on a parameter
type: Item
instance of: scholarly article
title: Tightness and continuity of a family of invariant measures for Markov chains depending on a parameter
zbMATH Open document ID: 0883.60064
DOI: 10.1007/BF02104664
published in: Siberian Mathematical Journal
publication date: 1998-03-16T00:00:00Z
review text: A family of Markov chains with transition functions \((P^{(\delta)}(x,B))_{\delta\in R}\) (\(x\in S\): metric space, \(B\in{\mathcal B}(S)\)) is considered. Every kernel \(p^{(\delta)}\) has at least one invariant distribution \(\pi^{(\delta)}\): \(\pi^{(\delta)}(B)= \int_S P^{(\delta)}(x,B)\pi^{(\delta)}(dx)\). Two conditions are proved to be sufficient for \(\pi^{(\delta)}\) to converge weakly to \(\pi^{(0)}\) \((\delta\to 0)\). They are the following: 1) for all \(x\in S\) under \(y\to x\), \(P^{(\delta)}(y,\cdot)\) tends weakly to \(P^{(0)}(x,\cdot)\), 2) the family \((\pi^{(\delta)})\) is tight in \(S\). For investigation of tightness of the family a method of Lyapunov test functions is used. These are some real functions with a special property of their increments on the chain. In case of \(S=R_+\) the tightness is connected with distributions of increments of the chain itself.
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 1023925
zbMATH Keywords: Markov kernel
zbMATH Keywords: weak convergence
zbMATH Keywords: Lyapunov test functions
zbMATH Keywords: increments
author: Dmitry Korshunov
reviewed by: B. P. Harlamov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1023925