Journal of Applied Probability

Spectral theory for weakly reversible Markov chains

Achim Wübker

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 49, Number 1 (2012), 245-265.

Dates
First available in Project Euclid: 8 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.jap/1331216845

Digital Object Identifier
doi:10.1239/jap/1331216845

Mathematical Reviews number (MathSciNet)
MR2952893

Zentralblatt MATH identifier
1246.37012

Subjects
Primary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Markov chain general state space spectral gap property isoperimetric constant reversibility

Citation

Wübker, Achim. Spectral theory for weakly reversible Markov chains. J. Appl. Probab. 49 (2012), no. 1, 245--265. doi:10.1239/jap/1331216845. https://projecteuclid.org/euclid.jap/1331216845


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