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A Markov chain associated with the minimal quasi-stationary distribution of birth–death chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Servet Martínez  and
Maria Eulália Vares
Servet Martínez*
Affiliation:
Universidad de Chile
Maria Eulália Vares*
Affiliation:
Instituto de Matemática Pura e Aplicada, Rio de Janeiro
*
∗Postal address: Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170–3 Correo 3, Santiago, Chile.
∗∗Postal address: Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardîm Botânico, 22460 Rio de Janeiro, Brasil.
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Abstract

We show that if the limiting conditional distribution for an absorbed birth–death chain exists, then the chain conditioned to non-absorption converges to a Markov chain with transition probabilities given by the matrix associated with the minimal quasi-stationary distribution.

Keywords

YAGLOM LIMITRANDOM WALKS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 25 - 38
Copyright
Copyright © Applied Probability Trust 1995 

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References

[1] Ferrari, P., Martínez, S. and Picco, P. (1991) Some properties of quasi-stationary distributions in the birth-death chains: a dynamical approach. In Instabilities and Non-Equilibrium Structures III, pp. 177–187. Kluwer, Dordrecht.Google Scholar
[2] Ferrari, P., Martínez, S. and Picco, P. (1992) Existence of quasi-stationary distributions for birth-death chains. Adv. Appl. Prob. 24, 795–813.CrossRefGoogle Scholar
[3] Ferrari, P., Kesten, H., Martínez, S. and Picco, P. (1993) Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Prob. To appear.Google Scholar
[4] Keener, R. W. (1992) Limit theorems for random walks conditioned to stay positive. Ann. Prob. 20, 801–824.Google Scholar
[5] Kesten, H. (1995) A ratio limit theorem for (sub)Markov chains on {1, 2, …} with bounded jumps. Adv. Appl. Prob. 27(3).Google Scholar
[6] Martínez, S. (1993) Quasi-stationary distributions for birth-death chains, cellular automata and cooperative systems. NATO ASI Series, pp. 491–505. Kluwer, Dordrecht.Google Scholar
[7] Pakes, A. G. (1995) Quasi-stationary laws for Markov processes: examples of an always approximate absorbing state. Adv. Appl. Prob. 27, 120–145.Google Scholar
[8] Roberts, G. O. and Jacka, S. D. (1994) Weak convergence of conditioned birth and death processes. J. Appl. Prob. 31, 90–100.Google Scholar
[9] Seneta, E. (1973) Non-Negative Matrices and Markov Chains. Springer-Verlag, New York.Google Scholar
[10] 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, 404–434.Google Scholar
[11] Van Doorn, E. and Schrijner, P. (1992) Geometric ergodicity, quasi-stationary and ratio limits for random walks. Preprint, University of Twente.Google Scholar
[12] Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford (2)13, 7–28.Google Scholar