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Uniqueness criteria for continuous-time Markov chains with general transition structures

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Anyue Chen ,
Phil Pollett ,
Hanjun Zhang  and
Ben Cairns
Anyue Chen*
Affiliation:
University of Greenwich and the University of Hong Kong
Phil Pollett*
Affiliation:
The University of Queensland
Hanjun Zhang*
Affiliation:
The University of Queensland
Ben Cairns*
Affiliation:
The University of Queensland
*
∗ Current address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Email address: a.chen@hknstasc.hku.hk
∗∗ Postal address: Department of Mathematics, The University of Queensland, Qld 4072, Australia.
∗∗ Postal address: Department of Mathematics, The University of Queensland, Qld 4072, Australia.
∗∗∗∗∗ Current address: School of Biological Sciences, University of Bristol, Woodland Road, Clifton, Bristol BS8 1UG, UK. Email address: Ben.Cairns@bristol.ac.uk
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Abstract

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We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

Keywords

Upwardly skip-free processdownwardly skip-free processMarkov branching processbirth-death process

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 1056 - 1074
Copyright
Copyright © Applied Probability Trust 2005 

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