Fluctuation Theorems for Synchronization of Interacting Polya's urns

We consider a model of N two-colors urns in which the reinforcement of each urn depends also on the content of all the other urns. This interaction is of mean-field type and it is tuned by a parameter $\alpha$ in [0,1]; in particular, for $\alpha=0$ the N urns behave as N independent Polya's urns. As shown in [9], for $\alpha>0$ urns synchronize, in the sense that the fraction of balls of a given color converges a.s. to the same (random) limit in all urns. In this paper we study fluctuations around this synchronized regime. The scaling of these fluctuations depends on the parameter $\alpha$. In particular the standard scaling $t^{-1/2}$ appears only for $\alpha>1/2$. For $\alpha\geq 1/2$ we also determine the limit distribution of the rescaled fluctuations. We use the notion of stable convergence, which is stronger than convergence in distribution.