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Weak convergence of conditioned birth and death processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

G. O. Roberts  and
S. D. Jacka
G. O. Roberts*
Affiliation:
University of Cambridge
S. D. Jacka*
Affiliation:
University of Warwick
*
∗ Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK.
∗∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
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Abstract

We consider the problem of conditioning a non-explosive birth and death process to remain positive until time T, and consider weak convergence of this conditional process as T → ∞. By a suitable almost sure construction we prove weak convergence. The almost sure construction used is of independent interest but relies heavily on the strong monotonic properties of birth and death processes.

Keywords

CONDITIONED PROCESSSTRONG MONOTONICITYNON-EXPLOSIVENESS

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 90 - 100
Copyright
Copyright © Applied Probability Trust 1994 

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