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).
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