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Existence of quasi-stationary distributions: A renewal dynamical approach. (English) Zbl 0827.60061

Summary: 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 \(x \to \infty\), we show that the existence of a qsd is equivalent to \(E_x e^{\lambda R} < \infty\) for some positive \(\lambda\) and \(x\). We also prove that a subsequence of \(\Phi^n \delta_x\) converges to a minimal qsd. For a birth and death process we prove that \(\Phi^n \delta_x\) converges along the full sequence to the minimal 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 \(\Phi \mu\).

MSC:

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