Random replacements in P\'olya urns with infinitely many colours

TL;DR

This paper extends the theory of Pólya urns with infinitely many colours to include random replacements, showing they can be represented as urns with deterministic replacements on an expanded colour space.

Contribution

It introduces a novel approach to model Pólya urns with random replacements by transforming them into deterministic urns on an augmented colour space.

Findings

Random replacements can be represented as deterministic urns on an extended colour space.

The framework generalizes previous models to arbitrary Borel spaces.

Provides a unified approach to analyze complex Pólya urns.

Abstract

We consider the general version of P\'olya 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 $S\times[0,1]$.