Polya urns via the contraction method

TL;DR

This paper introduces a novel approach using the contraction method and a combinatorial embedding into random trees to analyze the asymptotic behavior of Pólya urns, revealing various limit distributions.

Contribution

It develops a new combinatorial discrete-time embedding and applies the contraction method to derive asymptotic distributions of Pólya urns, including normal, non-normal, and periodic laws.

Findings

Limit laws with normal distribution

Limit laws with non-normal distribution

Asymptotic periodic distributional behavior

Abstract

We propose an approach to analyze the asymptotic behavior of P\'olya urns based on the contraction method. For this, a new combinatorial discrete time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each color. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete P\'olya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behavior.