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

DOI10.1239/JAP/1409932672zbMATH Open1326.37005arXiv1202.0677OpenAlexW2000233251MaRDI QIDQ2923434

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Publication date: 15 October 2014

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Abstract: We study the long time behaviour of a Markov process evolving in $mathbbN$ and conditioned not to hit 0. Assuming that the process comes back quickly from infinity, we prove that the process admits a unique quasi-stationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to infinity). Moreover, we prove that the distribution of the process converges exponentially fast in total variation norm to its quasi-stationary distribution and we provide an explicit rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasi-stationary distribution for birth and death processes: we prove that these processes converge exponentially fast to a quasi-stationary distribution if and only if they have a unique quasi-stationary distribution. Also, considering the lack of results on quasi-stationary distributions for non-irreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasi-stationary distribution for a class of possibly non-irreducible processes.

Full work available at URL: https://arxiv.org/abs/1202.0677

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