A New Approach to P\'olya Urn Schemes and Its Infinite Color Generalization

TL;DR

This paper introduces a novel approach to generalized Polya urn schemes with infinitely many colors, using branching Markov chains on random recursion trees to derive broad asymptotic results and extend classical finite urn findings.

Contribution

It presents a new method for analyzing infinite color Polya urn schemes via branching Markov chains, enabling general asymptotic analysis and new results.

Findings

Derived general asymptotics for infinite color urns

Unified analysis of classical finite urn results

Generated new results for infinite color urn models

Abstract

In this work we generalize Polya urn schemes with possibly infinitely many colors and extend the earlier models described in [4, 5, 7]. We provide a novel and unique approach of representing the observed sequence of colors in terms a branching Markov chain on random recursion tree. This enables us to derive fairly general asymptotic for our urn schemes. We then illustrate through several examples that our method can easily derive the classical results for finite urns, as well as, many new results for infinite color urns.