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Uniqueness criteria for continuous-time Markov chains with general transition structures. (English) Zbl 1101.60055

The main result of the paper is the following generalization of G. E. H. Reuter’s lemma [Acta Math. 97, 1–46 (1957; Zbl 0079.34703)]. Let {σ n :n0} be a sequence of real numbers satisfying 0σ 0 <σ 1 and

σ n+1 -σ n =f n σ n +h n + m=1 n g nm (σ m -σ m-1 ),n1,

where {f n :n1}, {h n :n1}, and {g nm :n1,1mn} are all nonnegative. Then {σ n } is bounded if and only if n=1 R n <, where {R n :n1} is defined recursively by R 1 =r 1 and for n2, R n =r n + m=2 n g nm R m-1 with r n =f n +h n +g n1 , n1·

As an application the authors give an alternative proof of a special case of Theorem 1.1 of [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.

60J27Continuous-time Markov processes on discrete state spaces
60J35Transition functions, generators, resolvents