SYNCHRONIZATION AND FLUCTUATIONS FOR INTERACTING STOCHASTIC SYSTEMS WITH INDIVIDUAL AND COLLECTIVE REINFORCEMENT
Résumé
The Pólya urn is the paradigmatic example of a reinforced stochastic process. It leads to a random time-limit. The Friedman urn is a generalization whose a.s. time-limit is not random anymore. In this work, following previous recent works, we introduce a new family of (finite) systems of reinforced stochastic processes, interacting through an additive collective reinforcement of mean field type. The two reinforcement rules strengths are tuned through different rates n^−γ. In the case the reinforcement rates are like n −1 , these reinforcements are of Pólya or Friedman type and may thus lead to random limits or not. Different parameter regimes needs to be considered. We state two kind of results. First, we study the time-asymptotics and show that L2 and almost sure convergence always holds. In particular, all the components share the same time-limit. We show, the nature of the limit (random/deterministic) depends on the parameters' regime considered. Second, we study fluctuations and prove central limit theorems whose scaling coefficient vary according to the regime considered.
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