Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]