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April, 1995 Existence of Quasi-Stationary Distributions. A Renewal Dynamical Approach
P. A. Ferrari, H. Kesten, S. Martinez, P. Picco
Ann. Probab. 23(2): 501-521 (April, 1995). DOI: 10.1214/aop/1176988277


We consider Markov processes on the positive integers for which the origin is an absorbing state. Quasi-stationary distributions (qsd's) are described as fixed points of a transformation $\Phi$ in the space of probability measures. Under the assumption that the absorption time at the origin, $R,$ of the process starting from state $x$ goes to infinity in probability as $x \rightarrow \infty$, we show that the existence of a $\operatorname{qsd}$ is equivalent to $E_xe^{\lambda R} < \infty$ for some positive $\lambda$ and $x$. We also prove that a subsequence of $\Phi^n\delta_x$ converges to a minimal $\operatorname{qsd}$. For a birth and death process we prove that $\Phi^n\delta_x$ converges along the full sequence to the minimal $\operatorname{qsd}$. The method is based on the study of the renewal process with interarrival times distributed as the absorption time of the Markov process with a given initial measure $\mu$. The key tool is the fact that the residual time in that renewal process has as stationary distribution the distribution of the absorption time of $\Phi\mu$.


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P. A. Ferrari. H. Kesten. S. Martinez. P. Picco. "Existence of Quasi-Stationary Distributions. A Renewal Dynamical Approach." Ann. Probab. 23 (2) 501 - 521, April, 1995.


Published: April, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0827.60061
MathSciNet: MR1334159
Digital Object Identifier: 10.1214/aop/1176988277

Primary: 60J27
Secondary: 60J80 , 60K05

Keywords: Quasi-stationary distributions , renewal processes , residual time

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 2 • April, 1995
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