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Existence and Uniqueness of a Quasistationary Distribution for Markov Processes with Fast Return from Infinity

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

Published online by Cambridge University Press:  30 January 2018

Servet Martínez ,
Jaime San Martín  and
Denis Villemonais
Servet Martínez*
Affiliation:
Universidad de Chile
Jaime San Martín*
Affiliation:
Universidad de Chile
Denis Villemonais*
Affiliation:
Université de Lorraine
*
∗ Postal address: Departemento Ingeniería, Matemática and Centro de Modelamiento Matemático, Universidad de Chile, UMI 2807, CNRS, Universidad de Chile, Santiago, Chile.
∗∗ Postal address: Departemento Ingeniería, Matemática and Centro de Modelamiento Matemático, Universidad de Chile, UMI 2807, CNRS, Universidad de Chile, Santiago, Chile.
∗∗∗ Postal address: Institut Élie Cartan de Lorraine, Université de Lorraine, Site de Nancy, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex, France. Email address: denis.villemonais@inria.fr
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Abstract

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We study the long-time behaviour of a Markov process evolving in N and conditioned not to hit 0. Assuming that the process comes back quickly from ∞, we prove that the process admits a unique quasistationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to ∞). Moreover, we prove that the distribution of the process converges exponentially fast in the total variation norm to its quasistationary distribution and we provide a bound for the rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasistationary distribution for birth-and-death processes: we prove that starting from any initial distribution the conditional probability converges in law to a unique distribution ρ supported in N* if and only if the process has a unique quasistationary distribution. Moreover, ρ is this unique quasistationary distribution and the convergence is shown to be exponentially fast in the total variation norm. Also, considering the lack of results on quasistationary distributions for nonirreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasistationary distribution for a class of possibly nonirreducible processes.

Keywords

Process with absorptionquasistationary distributionYaglom limitmixing propertybirth-and-death process

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing 60B10: Convergence of probability measures 60F99: None of the above, but in this section
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 756 - 768
Copyright
© Applied Probability Trust 

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