P\'olya Urn Schemes with Infinitely Many Colors

TL;DR

This paper introduces a new infinite-color urn model based on lattice-indexed colors, proving Gaussian limit theorems for the color distribution of the nth ball, regardless of the underlying walk's recurrence properties.

Contribution

It develops a novel representation of the urn's color as a sampled point on a random walk path, enabling new limit theorems for infinite-color urns.

Findings

Asymptotic distribution is Gaussian after centering and scaling.

Order of centering is logarithmic in n.

Order of scaling is square root of logarithm in n.

Abstract

In this work we introduce a new type of urn model with infinite but countable many colors indexed by an appropriate infinite set. We mainly consider the indexing set of colors to be the $d$-dimensional integer lattice and consider balanced replacement schemes associated with bounded increment random walks on it. We prove central and local limit theorems for the random color of the $n$-th selected ball and show that irrespective of the null recurrent or transient behavior of the underlying random walks, the asymptotic distribution is Gaussian after appropriate centering and scaling. We show that the order of any non-zero centering is always ${\mathcal O}\left(\log n\right)$ and the scaling is ${\mathcal O}\left(\sqrt{\log n}\right)$. The work also provides similar results for urn models with infinitely many colors indexed by more general lattices in ${\mathbb R}^d$. We introduce a novel technique of representing the random color of the $n$-th selected ball as a suitably sampled point on the path of the underlying random walk. This helps us to derive the central and local limit theorems.