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

A Chen, PK Pollett, H Zhang, BJ Cairns

Research output: Contribution to journalArticle (Academic Journal)peer-review

18 Citations (Scopus)

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 Reuter (1957), 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 (1999) 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.
Translated title of the contributionUniqueness criteria for continuous-time Markov chains with general transition structure
Original languageEnglish
Pages (from-to)1056 - 1074
Number of pages19
JournalAdvances in Applied Probability
Volume37
DOIs
Publication statusPublished - Dec 2005

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