Infinite-color randomly reinforced urns with dominant colors

TL;DR

This paper studies a class of reinforced urn processes with dominant colors, proving almost sure convergence of distributions and exploring implications for Bayesian inference, clinical trials, and species sampling.

Contribution

It introduces and analyzes dominant Pólya sequences with unbounded colors, establishing convergence results and rates, and discusses potential applications.

Findings

Predictive and empirical distributions converge almost surely to a random measure.

Dominant colors with maximum expected reinforcement dominate the process.

Convergence rates are provided for the distributions.

Abstract

We define and prove limit results for a class of dominant P\'olya sequences, which are randomly reinforced urn processes with color-specific random weights and unbounded number of possible colors. Under fairly mild assumptions on the expected reinforcement, we show that the predictive and the empirical distributions converge almost surely (a.s.) in total variation to the same random probability measure $\tilde{P}$; moreover, $\tilde{P}(\mathcal{D})=1$ a.s., where $\mathcal{D}$ denotes the set of dominant colors for which the expected reinforcement is maximum. In the general case, the predictive probabilities and the empirical frequencies of any $\delta$-neighborhood of $\mathcal{D}$ converge a.s. to one. That is, although non-dominant colors continue to be regularly observed, their distance to $\mathcal{D}$ converges in probability to zero. We refine the above results with rates of convergence. We further hint potential applications of dominant P\'olya sequences in randomized clinical trials and species sampling, and use our central limit results for Bayesian inference.