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Zbl 1101.60055
Chen, Anyue; Pollett, Phil; Zhang, Hanjun; Cairns, Ben
Uniqueness criteria for continuous-time Markov chains with general transition structures.
(English)
[J] Adv. Appl. Probab. 37, No. 4, 1056-1074 (2005). ISSN 0001-8678

The main result of the paper is the following generalization of {\it G. E. H. Reuter}'s lemma [Acta Math. 97, 1--46 (1957; Zbl 0079.34703)]. Let $\{\sigma_n:n\geq 0\}$ be a sequence of real numbers satisfying $0\leq\sigma_0<\sigma_1$ and $$\sigma_{n+1}-\sigma_n=f_n\sigma_n+h_n+\sum_{m=1}^ng_{nm}(\sigma_m-\sigma_{m-1}),\;n\geq 1,$$ where $\{f_n:n\geq 1\},$ $\{h_n:n\geq 1\},$ and $\{g_{nm}:n\geq 1,\,1\leq m\leq n\}$ are all nonnegative. Then $\{\sigma_n\}$ is bounded if and only if $\sum_{n=1}^\infty R_n<\infty,$ where $\{R_n:n\geq 1\}$ is defined recursively by $R_1=r_1$ and for $n\geq 2,$ $R_n=r_n+\sum_{m=2}^ng_{nm}R_{m-1}$ with $r_n=f_n+h_n+g_{n1},$ $n\geq 1.$ \par As an application the authors give an alternative proof of a special case of Theorem 1.1 of [{\it M. Chen}, Chin. Ann. Math., Ser. B 20, No.~1, 77--82 (1999; Zbl 0922.60068)] concerning upwardly skip-free processes. The authors use their generalization of Reuter's lemma and obtain some new results for downwardly skip-free chains, such as Markov branching processes. Finally, they study asymptotic birth-death processes being neither upwardly nor downwardly skip-free.
[Roman Urban (Wrocław)]
MSC 2000:
*60J27 Markov chains with continuous parameter
60J35 Transition functions

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

Citations: Zbl 0079.34703; Zbl 0922.60068

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