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Comments on the age distribution of Markov processes

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes  and
Simon Tavaré
Anthony G. Pakes*
Affiliation:
University of Western Australia
Simon Tavaré
Affiliation:
University of Utah
*
∗Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA6009, Australia.
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Abstract

Previous work on the concept of a limiting conditional age distribution of a discrete-state continuous-time Markov process with one absorbing state is generalised. The generalisation allows this process to have a finite number of absorbing states and the associated return process to have an arbitrary initial distribution on the transient states of the absorbing process. If the return process is ρ-recurrent, possesses the strong ratio limit property and satisfies some further requirements then the limiting age distribution exists. The proof of this result requires a new representation of the ρ-invariant measure of the return process.

The following examples are treated, (a) finite state space birth-death processes, (b) Markov branching processes and the linear death process, and (c) the linear birth and death process with killing.

Keywords

MARKOV PROCESSLIMITING AGEρ-CLASSIFICATIONSTRONG RATIO LIMIT PROPERTYBIRTH AND DEATH PROCESSESMARKOV BRANCHING PROCESSESLIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 4 , December 1981 , pp. 681 - 703
Copyright
Copyright © Applied Probability Trust 1981 

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Footnotes

a

Present address: Department of Statistics, Colorado State University, Fort Collins, Colorado, 80523, U.S.A.

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