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Weak convergence of conditioned processes on a countable state space

Part of: Markov processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

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

We consider the problem of conditioning a continuous-time Markov chain (on a countably infinite state space) not to hit an absorbing barrier before time T; and the weak convergence of this conditional process as T → ∞. We prove a characterization of convergence in terms of the distribution of the process at some arbitrary positive time, t, introduce a decay parameter for the time to absorption, give an example where weak convergence fails, and give sufficient conditions for weak convergence in terms of the existence of a quasi-stationary limit, and a recurrence property of the original process.

Keywords

MARKOV CHAINEVANESCENT CHAINQUASI-STATIONARY DISTRIBUTION

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60B10: Convergence of probability measures
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 902 - 916
Copyright
Copyright © Applied Probability Trust 1995 

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