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Spectral Theory for Weakly Reversible Markov Chains

Part of: Markov processes Ergodic theory

Published online by Cambridge University Press:  04 February 2016

Achim Wübker
Achim Wübker*
Affiliation:
Universität Osnabrück
*
∗ Postal address: Fachbereich Mathematik, Universität Osnabrück, Albrechtstrasse 28a, 49076 Osnabrück, Germany. Email address: [email protected]
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Abstract

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The theory of L2-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility with a weaker assumption, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of the isoperimetric constant. We show that this result can be applied to a large class of Markov chains, including those that are related to positive recurrent finite-range random walks on Z.

Keywords

Markov chaingeneral state spacespectral gap propertyisoperimetric constantreversibility

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 245 - 265
Copyright
© Applied Probability Trust 

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