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Markovian bounds on functions of finite Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

James Ledoux  and
Laurent Truffet
James Ledoux*
Affiliation:
Institut National des Sciences Appliquées, Rennes
Laurent Truffet*
Affiliation:
Institut de Recherche en Communications et Cybernetique de Nantes
*
∗ Postal address: INSA, 20 avenue des Buttes de Coesmes, 35043 Rennes Cedex, France. Email address: james.ledoux@insa.rennes.fr
∗∗ Postal address: Ecole des Mines de Nantes, Département Automatique–Productique, 4 rue Alfred Kastler BP 20722, 44307 Nantes Cedex 3, France.
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Abstract

In this paper, we obtain Markovian bounds on a function of a homogeneous discrete time Markov chain. For deriving such bounds, we use well-known results on stochastic majorization of Markov chains and the Rogers–Pitman lumpability criterion. The proposed method of comparison between functions of Markov chains is not equivalent to generalized coupling method of Markov chains, although we obtain same kind of majorization. We derive necessary and sufficient conditions for existence of our Markovian bounds. We also discuss the choice of the geometric invariant related to the lumpability condition that we use.

Keywords

weak lumpabilitystochastic comparisonstrong ordering

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 2 , June 2001 , pp. 505 - 519
Copyright
Copyright © Applied Probability Trust 2001 

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