• Bernoulli
  • Volume 23, Number 4B (2017), 3243-3267.

Pólya urn schemes with infinitely many colors

Antar Bandyopadhyay and Debleena Thacker

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In this work, we introduce a class of balanced urn schemes with infinitely many colors indexed by ${\mathbb{Z} }^{d}$, where the replacement schemes are given by the transition matrices associated with bounded increment random walks. We show that the color of the $n$th selected ball follows a Gaussian distribution on ${\mathbb{R} }^{d}$ after ${\mathcal{O} }(\log n)$ centering and ${\mathcal{O} }(\sqrt{\log n})$ scaling irrespective of whether the underlying walk is null recurrent or transient. We also provide finer asymptotic similar to local limit theorems for the expected configuration of the urn. The proofs are based on a novel representation of the color of the $n$th selected ball as “slowed down” version of the underlying random walk.

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Bernoulli, Volume 23, Number 4B (2017), 3243-3267.

Received: July 2015
Revised: February 2016
First available in Project Euclid: 23 May 2017

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central limit theorem infinite color urn local limit theorem random walk reinforcement processes urn models


Bandyopadhyay, Antar; Thacker, Debleena. Pólya urn schemes with infinitely many colors. Bernoulli 23 (2017), no. 4B, 3243--3267. doi:10.3150/16-BEJ844.

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