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

Abstract

We study the long-time behaviour of a Markov process evolving in $\mathbb{N}$ and conditioned not to hit 0. Assuming that the process comes back quickly from $\infty$, 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 $\infty$). 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 $\rho$ supported in $\mathbb{N}^*$ if and only if the process has a unique quasistationary distribution. Moreover, $\rho$ 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.