Journal of Applied Probability

Spectral theory for weakly reversible Markov chains

Achim Wübker

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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.

Article information

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

First available in Project Euclid: 8 March 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

Markov chain general state space spectral gap property isoperimetric constant reversibility


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

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