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Limiting Conditional Distributions for Birthdeath Processes

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

M. Kijima ,
M. G. Nair ,
P. K. Pollett  and
E. A. Van Doorn
M. Kijima*
Affiliation:
University of Tsukuba
M. G. Nair*
Affiliation:
Curtin University of Technology
P. K. Pollett*
Affiliation:
University of Queensland
E. A. Van Doorn*
Affiliation:
University of Twente
*
∗ Postal address: Graduate School of Systems Management, University of Tsukuba, Tokyo 112, Japan.
∗∗ Postal address: School of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6001, Australia.
∗∗∗ Postal address: Department of Mathematics, University of Queensland, QLD 4072, Australia.
∗∗∗∗ Postal address: Faculty of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands.
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Abstract

In a recent paper [16], one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

Keywords

INVARIANT MEASURESQUASI-STATIONARY DISTRIBUTIONS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 185 - 204
Copyright
Copyright © Applied Probability Trust 1997 

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