Synchronization and fluctuation theorems for interacting Friedman urns

Abstract

We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for fluctuations around the synchronization limit.