Journal of Theoretical Probability

, Volume 26, Issue 1, pp 198–221 | Cite as

Asymptotic Optimality of Isoperimetric Constants

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Abstract

In this paper, we investigate the existence of L 2(π)-spectral gaps for π-irreducible, positive recurrent Markov chains with a general state space Ω. We obtain necessary and sufficient conditions for the existence of L 2(π)-spectral gaps in terms of a sequence of isoperimetric constants. For reversible Markov chains, it turns out that the spectral gap can be understood in terms of convergence of an induced probability flow to the uniform flow. These results are used to recover classical results concerning uniform ergodicity and the spectral gap property as well as other new results. As an application of our result, we present a rather short proof for the fact that geometric ergodicity implies the spectral gap property. Moreover, the main result of this paper suggests that sharp upper bounds for the spectral gap should be expected when evaluating the isoperimetric flow for certain sets. We provide several examples where the obtained upper bounds are exact.

Keywords

Markov chain Spectral gap Isoperimetric constants 

Mathematics Subject Classification (2000)

60J05 60J25 37A30 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität OsnabrückOsnabrückGermany

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