A stochastic approximation approach to quasi-stationary distributions on finite spaces

Michel Benaïm (1) , Bertrand Cloez (2, 3)

1 UNINE - Institut de Mathématiques
2 MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie
3 MODEMIC - Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme

This work is concerned with the analysis of a stochastic approximation algorithm for the simulation of quasi-stationary distributions on finite state spaces. This is a generalization of a method introduced by Aldous, Flannery and Palacios. It is shown that the asymptotic behavior of the empirical occupation measure of this process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the unit simplex. This approach provides new proof of convergence as well as precise rates for this type of algorithm. We then compare this algorithm with particle system algorithms.

Mots clés: reinforced random walks; random perturbations of dynamical systems; Quasi-stationary distributions; approximation method; random perturbations of dynamical systems.

Domaines: Probabilités [math.PR]

Michel Benaïm, Bertrand Cloez. A stochastic approximation approach to quasi-stationary distributions on finite spaces. 2014. ⟨hal-00986430v2⟩