Advances in Applied Probability

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

Anyue Chen, Phil Pollett, Hanjun Zhang, and Ben Cairns

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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.

Article information

Adv. in Appl. Probab. Volume 37, Number 4 (2005), 1056-1074.

First available in Project Euclid: 14 December 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Upwardly skip-free process downwardly skip-free process Markov branching process birth-death process


Chen, Anyue; Pollett, Phil; Zhang, Hanjun; Cairns, Ben. Uniqueness criteria for continuous-time Markov chains with general transition structures. Adv. in Appl. Probab. 37 (2005), no. 4, 1056--1074. doi:10.1239/aap/1134587753.

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