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

Authors: A. Chen, P. Pollett, H. Zhang and B. Cairns.

Reference: Advances in Applied Probability 37: 1056-1074.

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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.

Keywords. Upwardly skip-free process; downwardly skip-free process; Markov branching process; birth-death process.

Acknowledgement. We would like to thank the referee for valuable comments and suggestions, which lead to a much improved presentation of our results. The support of the Australian Research Council (grant no. A00104575) is gratefully acknowledged. The work of Ben Cairns is supported by a PhD scholarship from the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems

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Cancer Epidemiology Unit, University of Oxford, Richard Doll Building, Roosevelt Drive, Oxford OX3 7LF, U.K.