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Quasistationarity of continuous-time Markov chains with positive drift

Published online by Cambridge University Press:  17 February 2009

Pauline Coolen-Schrijner ,
Andrew Hart  and
Phil Pollett
Pauline Coolen-Schrijner
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, UK.
Andrew Hart
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld 4001, Australia.
Phil Pollett
Affiliation:
Department of Mathematics, The University of Queensland, Qld 4072, Australia.
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Abstract

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We shall study continuous-time Markov chains on the nonnegative integers which are both irreducible and transient, and which exhibit discernible stationarity before drift to infinity “sets in”. We will show how this ‘quasi’ stationary behaviour can be modelled using a limiting conditional distribution: specifically, the limiting state probabilities conditional on not having left 0 for the last time. By way of a dual chain, obtained by killing the original process on last exit from 0, we invoke the theory of quasistationarity for absorbing Markov chains. We prove that the conditioned state probabilities of the original chain are equal to the state probabilities of its dual conditioned on non-absorption, thus allowing to establish the simultaneous existence and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always quasistationary distribution in the usual sense, a similar statement is not possible for the original chain.

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 4 , April 2000 , pp. 423 - 441
Copyright
Copyright © Australian Mathematical Society 2000

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