Urn models with two types of strategies

TL;DR

This paper analyzes an urn model with two reinforcement strategies, deriving its long-term behavior, including laws of large numbers, central limit theorem, and phase transition phenomena.

Contribution

It introduces a mixed urn process combining Pólya-type and i.i.d. strategies, providing new asymptotic results and phase transition analysis.

Findings

Proves a law of large numbers for the urn proportion

Establishes a central limit theorem for fluctuations

Identifies a phase transition in the model behavior

Abstract

We study an urn process containing red and blue balls and two different strategies to reinforce the urn. Namely, a generalized P\'olya-type strategy versus an i.i.d. one. At each step, one of the two reinforcement strategies is chosen by flipping a coin. We study the asymptotic behaviour of this urn model, and prove a law of large numbers, a central limit theorem and a functional limit theorem for the proportion of balls into the urn. A phase transition is also stated.