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Geometric inequalities for the eigenvalues of concentrated Markov chains

Part of: Basic linear algebra Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Olivier François
Olivier François*
Affiliation:
LMC/IMAG
*
∗Postal address: LMC/IMAG, BP 53, 38041 Grenoble cedex 9, France. Email address: olivier.francois@imag.fr
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Abstract

This article describes new estimates for the second largest eigenvalue in absolute value of reversible and ergodic Markov chains on finite state spaces. These estimates apply when the stationary distribution assigns a probability higher than 0.702 to some given state of the chain. Geometric tools are used. The bounds mainly involve the isoperimetric constant of the chain, and hence generalize famous results obtained for the second eigenvalue. Comparison estimates are also established, using the isoperimetric constant of a reference chain. These results apply to the Metropolis-Hastings algorithm in order to solve minimization problems, when the probability of obtaining the solution from the algorithm can be chosen beforehand. For these dynamics, robust bounds are obtained at moderate levels of concentration.

Keywords

Reversible Markov chaineigenvaluesisoperimetric constantMetropolis-Hastings dynamicsminimization

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 65C05: Monte Carlo methods
Secondary: 15A42: Inequalities involving eigenvalues and eigenvectors
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 15 - 28
Copyright
Copyright © 2000 by The Applied Probability Trust 

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