The Annals of Probability

Existence of Quasi-Stationary Distributions. A Renewal Dynamical Approach

P. A. Ferrari, H. Kesten, S. Martinez, and P. Picco

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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 $\Phi\mu$.

Article information

Ann. Probab., Volume 23, Number 2 (1995), 501-521.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K05: Renewal theory

Quasi-stationary distributions renewal processes residual time


Ferrari, P. A.; Kesten, H.; Martinez, S.; Picco, P. Existence of Quasi-Stationary Distributions. A Renewal Dynamical Approach. Ann. Probab. 23 (1995), no. 2, 501--521. doi:10.1214/aop/1176988277.

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