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Existence of non-trivial quasi-stationary distributions in the birth-death chain

Part of: Convergence and divergence of infinite limiting processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Pablo A. Ferrari ,
Servet Martínez  and
Pierre Picco
Pablo A. Ferrari*
Affiliation:
Universidade de São Paulo
Servet Martínez*
Affiliation:
Universidad de Chile
Pierre Picco*
Affiliation:
Centre de Physique Théorique, CNRS-Luminy
*
∗Postal address: Instituto de Matemática e Estatistica, Universidade de São Paulo, Cx. Postal 20570, 01498 São Paulo, Brasil. E-mail: pablo@ime.usp.br
∗∗Postal address: Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Correo 3, Santiago 3, Chile. E-mail: smartine@uchcecvm.bitnet
∗∗∗Postal address: Laboratoire Propre, LP7061, Centre de Physique Théorique CNRS-Luminy, Case 907F, 13288 Marseille, Cedex 9, France. E-mail: picco@frcptm51.bitnet
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Abstract

We study conditions for the existence of non-trivial quasi-stationary distributions for the birth-and-death chain with 0 as absorbing state. We reduce our problem to a continued fractions one that can be solved by using extensions of classical results of this theory. We also prove that there exist normalized quasi-stationary distributions if and only if 0 is geometrically absorbing.

Keywords

MARKOV CHAINSNORMALIZED QUASI-STATIONARY DISTRIBUTIONBIRTH AND DEATH PROCESS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 40A15: Convergence and divergence of continued fractions
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 4 , December 1992 , pp. 795 - 813
Copyright
Copyright © Applied Probability Trust 1992 

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