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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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References

Darroch, J. H. and Seneta, E. (1967) On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196.CrossRefGoogle Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1993) Existence of quasi stationary distributions. A renewal dynamical approach. Universidade de São Paulo.Google Scholar
Jacka, S. D. and Roberts, G. O. (1992) Strong forms of weak convergence. University of Warwick, Department of Statistics Research Report No. 243.Google Scholar
Kesten, H. (1995) A ratio limit theorem for (sub) Markov chains on {1, 2, …} with bounded jumps. Adv. Appl. Prob. 27, 652–691.CrossRefGoogle Scholar
Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities, Proc. Lond. Math. Soc. (3) 13, 593–604.Google Scholar
Pinsky, R. G. (1985) On the convergence of diffusion processes conditioned to remain in a bounded region for a large time to limiting positive recurrent diffusion processes. Ann. Prob. 13, 363–378.Google Scholar
Roberts, G. O. (1991) Asymptotic approximations for Brownian motion boundary hitting times. Ann. Prob. 19, 1689–1731.CrossRefGoogle Scholar
Roberts, G. O. and Jacka, S. D. (1994) Weak convergence of conditioned birth and death processes. J. Appl. Prob. 31, 90–100.Google Scholar
Roberts, G. O., Jacka, S. D. and Pollett, P. K. (1994) Non-explosivity of limits of conditioned birth and death processes. Submitted.Google Scholar
Seneta, E. (1981) Non-negative Matrices and Markov Chains. Springer-Verlag, New York.Google Scholar
Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.Google Scholar