Random replacements in Pólya urns with infinitely many colours

Svante Janson

doi:10.1214/19-ecp226

Abstract

We consider the general version of Polya urns recently studied by Bandyopadhyay and Thacker (2016+) and Mailler and Marckert (2017), with the space of colours being any Borel space $S$ and the state of the urn being a finite measure on $S$. We consider urns with random replacements, and show that these can be regarded as urns with deterministic replacements using the colour space $S\times [0,1]$.

Highlights

We consider the general version of Pólya urns recently studied by Bandyopadhyay and Thacker (2016+) and Mailler and Marckert (2017), with the space of colours being any Borel space S and the state of the urn being a finite measure on S
The original Pólya urn, studied already in 1917 by Markov [14] but later named after Pólya who studied it in Eggenberger and Pólya [6] (1923) and Pólya [17] (1930), contains balls of two colours
The urn is a Markov process (Xn)∞ 0, with state space Z2 0. (The initial state X0 is some arbitrary given non-zero state.). This urn model has been generalized by various authors in a number of ways, all keeping the basic idea of a Markov process of sets of balls of different colours, where balls are drawn at random and the drawn balls determine the step in the process. (The extensions are all usually called Pólya urns, or perhaps generalized Pólya urns.) These generalizations have been studied by a large number of authors, and have found a large number of applications, see for example [12], [8], [3], [13] and the references given there

Summary

Introduction

The original Pólya urn, studied already in 1917 by Markov [14] but later named after Pólya who studied it in Eggenberger and Pólya [6] (1923) and Pólya [17] (1930), contains balls of two colours. The purpose of the present note is to show that this model with a measure-valued Pólya urn and the results for it by [3] and [13] extend almost automatically to the case of random replacements, at least in the case with no removals. Consider a measure-valued Pólya urn process (Xn)∞ 0 in a Borel space S, with random replacements. Many papers, including [3] and [13], consider only balanced Pólya urns, i.e., urns where the total number of balls added to the urn each time is deterministic, and the total number of balls in the urn after n steps is a deterministic linear function of n; in the measure-valued context, this means that the total mass Xn(S) = an + b, where b = X0(S).

Measurable spaces

Borel spaces

Two lemmas

Pólya urns

Deterministic replacements

Random replacement

Random replacements

An application