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Journal Articles The Annals of Applied Probability Year : 2018

Stochastic approximation of quasi-stationary distributions on compact spaces and applications

Abstract

In the continuity of a recent paper ([6]), dealing with finite Markov chains, this paper proposes and analyzes a recursive algorithm for the approximation of the quasi-stationary distribution of a general Markov chain living on a compact metric space killed in finite time. The idea is to run the process until extinction and then to bring it back to life at a position randomly chosen according to the (possibly weighted) empirical occupation measure of its past positions. General conditions are given ensuring the convergence of this measure to the quasi-stationary distribution of the chain. We then apply this method to the numerical approximation of the quasi-stationary distribution of a diffusion process killed on the boundary of a compact set and to the estimation of the spectral gap of irreducible Markov processes. Finally, the sharpness of the assumptions is illustrated through the study of the algorithm in a non-irreducible setting.
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Dates and versions

hal-01334603 , version 1 (21-06-2016)
hal-01334603 , version 2 (05-12-2016)
hal-01334603 , version 3 (06-11-2017)

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Cite

Michel Benaim, Bertrand Cloez, Fabien Panloup. Stochastic approximation of quasi-stationary distributions on compact spaces and applications. The Annals of Applied Probability, 2018, 28 (4), pp.2370-2416. ⟨10.1214/17-AAP1360⟩. ⟨hal-01334603v3⟩
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