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Subgeometric rates of convergence for a class of continuous-time Markov process

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Zhenting Hou ,
Yuanyuan Liu  and
Hanjun Zhang
Zhenting Hou*
Affiliation:
Central South University
Yuanyuan Liu*
Affiliation:
Central South University
Hanjun Zhang*
Affiliation:
University of Queensland
*
∗Postal address: School of Mathematics, Central South University, Changsha, Hunan, 410075, P. R. China.
∗Postal address: School of Mathematics, Central South University, Changsha, Hunan, 410075, P. R. China.
∗∗∗∗Postal address: Department of Mathematics, The University of Queensland, Queensland, 4072, Australia. Email address: hjz@maths.uq.edu.au
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Abstract

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Let (Φt)t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function Pt(x, ·) to π; specifically, we find conditions under which r(t)||Pt(x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

Keywords

Continuous-time Markov processqueueing modelbirth-death processergodicitysubgeometric convergence

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 698 - 712
Copyright
© Applied Probability Trust 2005 

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