Synchronization and fluctuations for interacting stochastic systems with individual and collective reinforcement

Abstract

The Pólya urn is the most representative example of a reinforced stochastic process. It leads to a random (non degenerated) time-limit. The Friedman urn is a natural generalization whose almost sure (a.s.) time-limit is not random any more. In this work, in the stream of previous recent works, we introduce a new family of (finite size) systems of reinforced stochastic processes, interacting through an additional collective reinforcement of mean field type. The two reinforcement rules strengths (one component-wise, one collective) are tuned through (possibly) two different rates. In special cases, these reinforcements are of Pólya or Friedman type as in urn contexts and may thus lead to limits which may be random or not. Different parameter regimes need to be considered. We state two kind of results. First, we study the time-asymptotic and show that L 2 and a.s. convergence always holds. Moreover, all the components share the same time-limit (so called synchronization phenomenon). We study the nature of the limit (random/deterministic) according to the parameters’ regime considered. Second, we study fluctuations by proving central limit theorems. Scaling coefficients vary according to the regime considered. This gives insights into many different rates of convergence. In particular, we identify the regimes where synchronization is faster than convergence toward the shared time-limit.