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On spectral gap estimates of a Markov chain via hitting times and coupling

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Christian Meise
Christian Meise*
Affiliation:
Universität Bielefeld
*
∗Postal address: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany. Email address: meise@mathematik.uni-bielefeld.de.
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Abstract

Well-known inequalities for the spectral gap of a discrete-time Markov chain, such as Poincaré's and Cheeger's inequalities, do not perform well if the transition graph of the Markov chain is strongly connected. For example in the case of nearest-neighbour random walk on the n-dimensional cube Poincaré's and Cheeger's inequalities are off by a factor n. Using a coupling technique and a contraction principle lower bounds on the spectral gap can be derived. Finally, we show that using the contraction principle yields a sharp estimate for nearest-neighbour random walk on the n-dimensional cube.

Keywords

Markov chainsspectral gapcouplings

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 310 - 319
Copyright
Copyright © Applied Probability Trust 1999 

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