Limit distributions for large P\'{o}lya urns

TL;DR

This paper analyzes the asymptotic behavior of large two-color Pólya urns, deriving explicit limit distributions for the scaled composition vector using complex analysis and differential equations.

Contribution

It introduces a novel approach to determine the limit distributions of large Pólya urns via embedding in continuous time and solving associated differential equations.

Findings

Explicit formulas for Fourier transforms of limit distributions.

Limit laws form a new family of probability densities on the real line.

Connection to Abelian integrals over Fermat curves.

Abstract

We consider a two-color P\'{o}lya urn in the case when a fixed number $S$ of balls is added at each step. Assume it is a large urn that is, the second eigenvalue $m$ of the replacement matrix satisfies $1/2<m/S\leq1$. After $n$ drawings, the composition vector has asymptotically a first deterministic term of order $n$ and a second random term of order $n^{m/S}$. The object of interest is the limit distribution of this random term. The method consists in embedding the discrete-time urn in continuous time, getting a two-type branching process. The dislocation equations associated with this process lead to a system of two differential equations satisfied by the Fourier transforms of the limit distributions. The resolution is carried out and it turns out that the Fourier transforms are explicitly related to Abelian integrals over the Fermat curve of degree $m$. The limit laws appear to constitute a new family of probability densities supported by the whole real line.