Existence of Quasi-Stationary Distributions. A Renewal Dynamical Approach

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

Ann. Probab., Tome 23 (1995) no. 3, p. 501-521 / Harvested from Project Euclid

Résumé

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 $\Phi\mu$.

Détail

Publié le : 1995-04-14
Classification:  Quasi-stationary distributions,  renewal processes,  residual time,  60J27,  60J80,  60K05