Journal of Applied Probability

Existence and uniqueness of a quasistationary distribution for Markov processes with fast return from infinity

Servet Martínez, Jaime San Martín, and Denis Villemonais

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 51, Number 3 (2014), 756-768.

Dates
First available in Project Euclid: 5 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1409932672

Digital Object Identifier
doi:10.1239/jap/1409932672

Mathematical Reviews number (MathSciNet)
MR3256225

Zentralblatt MATH identifier
1326.37005

Subjects
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.)

Keywords
Process with absorption quasistationary distribution Yaglom limit mixing property birth-and-death process

Citation

Martínez, Servet; San Martín, Jaime; Villemonais, Denis. Existence and uniqueness of a quasistationary distribution for Markov processes with fast return from infinity. J. Appl. Probab. 51 (2014), no. 3, 756--768. doi:10.1239/jap/1409932672. https://projecteuclid.org/euclid.jap/1409932672


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