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

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

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

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 asymptotic rates for this type of algorithm. In the last part, our convergence results are compared with those of a particle system algorithm (a discrete-time version of the Fleming-Viot algorithm).

Mots clés: approximation method; reinforced random; walks; random perturbations of dynamical

Mots clés: quasi-stationary distributions

Domaines: Probabilités [math.PR]

Michel Benaïm, Bertrand Cloez. A stochastic approximation approach to quasi-stationary distributions on finite spaces. Electronic Communications in Probability, 2015, 20 (37), pp.1-14. ⟨10.1214/ECP.v20-3956⟩. ⟨hal-01185942⟩