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November 2017 Pólya urn schemes with infinitely many colors
Antar Bandyopadhyay, Debleena Thacker
Bernoulli 23(4B): 3243-3267 (November 2017). DOI: 10.3150/16-BEJ844


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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Antar Bandyopadhyay. Debleena Thacker. "Pólya urn schemes with infinitely many colors." Bernoulli 23 (4B) 3243 - 3267, November 2017.


Received: 1 July 2015; Revised: 1 February 2016; Published: November 2017
First available in Project Euclid: 23 May 2017

zbMATH: 06778286
MathSciNet: MR3654806
Digital Object Identifier: 10.3150/16-BEJ844

Keywords: central limit theorem , infinite color urn , local limit theorem , Random walk , reinforcement processes , urn models

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability


Vol.23 • No. 4B • November 2017
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