Interacting generalized Friedman's urn systems

TL;DR

This paper analyzes systems of interacting Generalized Friedman's Urns with irreducible mean matrices, providing detailed asymptotic behavior, convergence rates, and CLTs using eigenanalysis and stochastic approximation.

Contribution

It introduces a comprehensive framework for understanding the asymptotic properties of interacting GFU systems, including subsystem behaviors and convergence characteristics.

Findings

Identification of subsystems with distinct behaviors

Limiting proportions and convergence rates established

Central Limit Theorems proved for urn proportions

Abstract

We consider systems of interacting Generalized Friedman's Urns (GFUs) having irreducible mean replacement matrices. The interaction is modeled through the probability to sample the colors from each urn, that is defined as convex combination of the urn proportions in the system. From the weights of these combinations we individuate subsystems of urns evolving with different behaviors. We provide a complete description of the asymptotic properties of urn proportions in each subsystem by establishing limiting proportions, convergence rates and Central Limit Theorems. The main proofs are based on a detailed eigenanalysis and stochastic approximation techniques.