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

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

Chen, Anyue, Pollett, Phil, Zhang, Hanjun and Cairns, Ben (2005) Uniqueness criteria for continuous-time Markov chains with general transition structures. Advances in Applied Probability, 37 (4). pp. 1056-1074. ISSN 0001-8678 (Print), 1475-6064 (Online) (doi:10.1239/aap/1134587753)

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Abstract

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.

Item Type: Article
Uncontrolled Keywords: uniqueness criteria, Markov chains
Subjects: Q Science > QA Mathematics
Pre-2014 Departments: School of Computing & Mathematical Sciences
Related URLs:
Last Modified: 14 Oct 2016 09:02
Selected for GREAT 2016: None
Selected for GREAT 2017: None
Selected for GREAT 2018: None
URI: http://gala.gre.ac.uk/id/eprint/961

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