Characterizations of asymptotic distributions of continuous-time P\'{o}lya processes

TL;DR

This paper introduces an elementary approach to analyze the asymptotic distributions of continuous-time Pólya processes, revealing gamma distribution limits for certain urn models using PDEs and moments.

Contribution

It provides a novel, simplified method to characterize asymptotic distributions of continuous-time Pólya urns, including cases with randomized replacement matrices.

Findings

Limiting distribution of Bagchi-Pal urn is gamma.

Method applies to a broad class of balanced, tenable urns.

Results extend to urns with randomized replacement matrices.

Abstract

We propose an elementary but effective approach to studying a general class of Poissonized tenable and balanced urns on two colors. We characterize the asymptotic behavior of the process via a partial differential equation that governs the process, coupled with the method of moments applied in a bootstrapped manner. We show that the limiting distribution of the process underlying the Bagchi-Pal urn is gamma. We also look into the tenable and balanced processes associated with randomized replacement matrix. Similar results carry over to the process, with minor modifications in the methods of proof, done mutatis mutandis.