## The Annals of Probability

### Central Limit Theorems for Infinite Urn Models

Michael Dutko

#### 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

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