A martingale approach for Pólya urn processes

Lucile Laulin

doi:10.1214/20-ecp321

Abstract

This paper is devoted to a direct martingale approach for Polya urn models asymptotic behaviour. A Polya process is said to be small when the ratio of its remplacement matrix eigenvalues is less than or equal to 1/2, otherwise it is called large. We find again some well-known results on the asymptotic behaviour for small and large urns processes. We also provide new almost sure properties for small urns processes.

Highlights

At the inital time n = 0, an urn is filled with α ≥ 0 red balls and β ≥ 0 white balls.at any time n ≥ 1 one ball is drawn randomly from the urn and its color observed.If it is red it is returned to the urn together with a additional red balls and b ≥ 0 white ones
This paper is devoted to a direct martingale approach for Pólya urn models asymptotic behaviour
We find again some well-known results on the asymptotic behaviour for small and large urn processes

Summary

Introduction

At any time n ≥ 1 one ball is drawn randomly from the urn and its color observed. If it is red it is returned to the urn together with a additional red balls and b ≥ 0 white ones. We establish the almost sure convergence and the asymptotic normality for this martingale.

Traditional Pólya urn model

Gereralized Pólya urn model

Critically small urns

Proofs of the almost sure convergence results

Generalized urn model – large urns

Traditional urn model

Generalized urn model – small urns

Generalized urn model – critically small urns