Stochastic Approximation, Cooperative Dynamics and Supermodular GamesJournal articleMathieu Faure and Michel Benaïm, Annals of Applied Probability, Volume 22, Issue 5, pp. 2133-2164, 2012

This paper considers a stochastic approximation algorithm, with decreasing step size and martingale difference noise. Under very mild assumptions, we prove the nonconvergence of this process toward a certain class of repulsive sets for the associated ordinary differential equation (ODE). We then use this result to derive the convergence of the process when the ODE is cooperative in the sense of Hirsch [SIAM J. Math. Anal. 16 (1985) 423-439]. In particular, this allows us to extend significantly the main result of Hofbauer and Sandholm [Econometrica 70 (2002) 2265-2294] on the convergence of stochastic fictitious play in supermodular games.

Stochastic Approximations of Set-Valued Dynamical Systems: Convergence with Positive Probability to an AttractorJournal articleMathieu Faure and Gregory Roth, Mathematics of Operations Research, Volume 35, Issue 3, pp. 624-640, 2010

A successful method to describe the asymptotic behavior of a discrete time stochastic process governed by some recursive formula is to relate it to the limit sets of a well-chosen mean differential equation. Under an attainability condition, Benaïm proved that convergence to a given attractor of the flow induced by this dynamical system occurs with positive probability for a class of Robbins Monro algorithms. Benaïm, Hofbauer, and Sorin generalised this approach for stochastic approximation algorithms whose average behavior is related to a differential inclusion instead. We pursue the analogy by extending to this setting the result of convergence with positive probability to an attractor.

Self-normalized Large Deviations for Markov ChainsJournal articleMathieu Faure, Electronic Journal of Probability, Volume 7, pp. 1-31, 2002
Principes de grandes déviations autonormalisées pour des chaînes de MarkovSelf-normalized large deviations for Markov chainsJournal articleMathieu Faure, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Volume 333, Issue 9, pp. 885-890, 2001

Nous prouvons un principe de grandes déviations autonormalisé pour la moyenne empirique de fonctionnelles additives non bornées d'une chaı̂ne de Markov. L'autonormalisation s'applique à des cas pour lesquels une hypothèse de domination serait nécessaire pour avoir un principe de grandes déviations traditionnel. Nous suivons ainsi la voie ouverte par Dembo et Shao [2] dans le cas de suites indépendantes et identiquement distribuées pour l'obtention de principes de grandes déviations partiels.