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A monotonicity in reversible Markov chains

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Robert Lund ,
Ying Zhao  and
Peter C. Kiessler
Robert Lund*
Affiliation:
Clemson University
Ying Zhao*
Affiliation:
The University of Georgia
Peter C. Kiessler*
Affiliation:
Clemson University
*
∗Postal address: Department of Mathematical Sciences, Clemson University, O-106 Martin Hall Box 340975, Clemson, SC 29634-0975, USA.
∗∗∗Postal address: Department of Statistics, The University of Georgia, Athens, GA 30602-1952, USA.
∗Postal address: Department of Mathematical Sciences, Clemson University, O-106 Martin Hall Box 340975, Clemson, SC 29634-0975, USA.
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Abstract

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In this paper we identify a monotonicity in all countable-state-space reversible Markov chains and examine several consequences of this structure. In particular, we show that the return times to every state in a reversible chain have a decreasing hazard rate on the subsequence of even times. This monotonicity is used to develop geometric convergence rate bounds for time-reversible Markov chains. Results relating the radius of convergence of the probability generating function of first return times to the chain's rate of convergence are presented. An effort is made to keep the exposition rudimentary.

Keywords

Convergence ratedecreasing hazard rateMarkov chainmonotonicityrenewal sequence

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 486 - 499
Copyright
© Applied Probability Trust 2006 

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