## Bernoulli

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

### Pólya urn schemes with infinitely many colors

#### Abstract

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.

#### Article information

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

Dates
Revised: February 2016
First available in Project Euclid: 23 May 2017

https://projecteuclid.org/euclid.bj/1495505092

Digital Object Identifier
doi:10.3150/16-BEJ844

Mathematical Reviews number (MathSciNet)
MR3654806

Zentralblatt MATH identifier
06778286

#### Citation

Bandyopadhyay, Antar; Thacker, Debleena. Pólya urn schemes with infinitely many colors. Bernoulli 23 (2017), no. 4B, 3243--3267. doi:10.3150/16-BEJ844. https://projecteuclid.org/euclid.bj/1495505092

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