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Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Servet Martínez  and
Bernard Ycart
Servet Martínez*
Affiliation:
Universidad de Chile
Bernard Ycart*
Affiliation:
Université René Descartes–Paris V
*
∗ Postal address: Centro Modelamiento Matemático, Universidad de Chile, UMR 2071-CNRS, Casilla 170/3, Santiago, Chile. Email address: smartine@dim.uchile.cl
∗∗ Postal address: Math–Info, 45 rue des Saints-Pères 75270, Paris Cedex 06, France.
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Abstract

For a positive recurrent continuous-time Markov chain on a countable state space, we compare the access time to equilibrium to the hitting time of a particular state. For monotone processes, the exponential rates are ranked. When the process starts far from equilibrium, a cutoff phenomenon occurs at the same instant, in the sense that both the access time to equilibrium and the hitting time of a fixed state are equivalent to the expectation of the latter. In the case of Markov chains on trees, that expectation can be computed explicitly. The results are illustrated on the M/M/∞ queue.

Keywords

cutoffgapconditionally invariant measures

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 188 - 205
Copyright
Copyright © Applied Probability Trust 2001 

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