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July, 1989 Central Limit Theorems for Infinite Urn Models
Michael Dutko
Ann. Probab. 17(3): 1255-1263 (July, 1989). DOI: 10.1214/aop/1176991268


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.


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Michael Dutko. "Central Limit Theorems for Infinite Urn Models." Ann. Probab. 17 (3) 1255 - 1263, July, 1989.


Published: July, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0685.60023
MathSciNet: MR1009456
Digital Object Identifier: 10.1214/aop/1176991268

Primary: 60F05
Secondary: 60C05

Keywords: central limit theorem , urn model

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 3 • July, 1989
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