The Annals of Probability

Central Limit Theorems for Infinite Urn Models

Michael Dutko

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Abstract

An urn model is defined as follows: $n$ balls are independently placed in an infinite set of urns and each ball has probability $p_k > 0$ of being assigned to the $k$th urn. We assume that $p_k \geq p_{k + 1}$ for all $k$ and that $\sum^\infty_{k = 1} p_k = 1$. A random variable $Z_n$ is defined to be the number of occupied urns after $n$ balls have been thrown. The main result is that $Z_n$, when normalized, converges in distribution to the standard normal distribution. Convergence to $N(0, 1)$ holds for all sequences $\{p_k\}$ such that $\lim_{n \rightarrow \infty} \operatorname{Var}Z_{N(n)} = \infty$, where $N(n)$ is a Poisson random variable with mean $n$. This generalizes a result of Karlin.

Article information

Source
Ann. Probab., Volume 17, Number 3 (1989), 1255-1263.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991268

Digital Object Identifier
doi:10.1214/aop/1176991268

Mathematical Reviews number (MathSciNet)
MR1009456

Zentralblatt MATH identifier
0685.60023

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

Keywords
Central limit theorem urn model

Citation

Dutko, Michael. Central Limit Theorems for Infinite Urn Models. Ann. Probab. 17 (1989), no. 3, 1255--1263. doi:10.1214/aop/1176991268. https://projecteuclid.org/euclid.aop/1176991268


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