Synchronization and fluctuation theorems for interacting Friedman urns. (English) Zbl 1356.60173

Sahasrabudhe, Neeraja

J. Appl. Probab. 53, No. 4, 1221-1239 (2016).

Summary: 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.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60F05	Central limit and other weak theorems
60B10	Convergence of probability measures
60F99	Limit theorems in probability theory

Keywords:

interacting urn model; Friedman urn; Pólya urn; synchronization; stable convergence; reinforcement; fluctuation theorem