A.s. convergence for infinite colour P\'olya urns associated with random walks

TL;DR

This paper proves almost sure convergence of the color distribution in infinite-color Pólya urns of a random walk type to a normal distribution, under minimal second moment assumptions, improving previous probabilistic convergence results.

Contribution

It establishes almost sure normal convergence for infinite-color Pólya urns with minimal second moment conditions, extending prior probabilistic convergence results.

Findings

Color distribution converges to a normal distribution almost surely.

Convergence holds under only second moment assumptions.

Results improve upon previous convergence in probability and exponential moment conditions.

Abstract

We consider P\'olya urns with infinitely many colours that are of a random walk type, in two related version. We show that the colour distribution a.s., after rescaling, converges to a normal distribution, assuming only second moments on the offset distribution. This improves results by Bandyopadhyay and Thacker (2014--2017; convergence in probability), and Mailler and Marckert (2017; a.s. convergence assuming exponential moment).