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

TL;DR

This paper analyzes a stochastic approximation algorithm for simulating quasi-stationary distributions on finite spaces, providing convergence proofs, rates, and comparisons with particle system methods.

Contribution

It extends a previous method by linking the empirical measure's behavior to a deterministic dynamical system, offering new convergence proofs and rate estimates.

Findings

Proves convergence of the algorithm to the quasi-stationary distribution

Provides explicit rates of convergence for the stochastic approximation

Compares the algorithm's performance with particle system algorithms

Abstract

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.