Stochastic approximation with random step sizes and urn models with random replacement matrices having finite mean. (English) Zbl 1435.62311

The article concerns a significant contribution to the stochastic approximation algorithm introduced by H. Robbins and S. Monro [Ann. Math. Stat. 22, 400–407 (1951; Zbl 0054.05901)]. The step sizes used in stochastic approximation are generally taken to be deterministic, and same is true for the drift (see [M. Benaïm, Lect. Notes Math. 1709, 1–68 (1999; Zbl 0955.62085); V. S. Borkar, Stochastic approximation. A dynamical systems viewpoint. Cambridge: Cambridge University Press; New Delhi: Hindustan Book Agency (2008; Zbl 1181.62119)]). The specific application of urn models (the original formulation due to [F. Eggenberger and G. Pólya, Z. Angew. Math. Mech. 3, 279–290 (1923; JFM 49.0382.01)]) with random replacement matrices needs to consider stochastic approximation in a setup where both the step sizes and the drift are random, but the sequence is uniformly bounded.
The paper deals with an extension to stochastic approximation algorithm for bounded sequences with random step size and drift. The result is applied to the stochastic approximation for an urn model with unbalanced random replacement matrix. It is shown that the corresponding differential equation is of Lotka-Volterra type which is able to analyze directly. It allows giving a complete analysis of urn models with balls of finitely many colors and random replacement matrix when it is assumed only that the first moment is finite.


62L20 Stochastic approximation
60F15 Strong limit theorems
60G42 Martingales with discrete parameter
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