A Berry-Esseen bound for the uniform multinomial occupancy model

Abstract

The inductive size bias coupling technique and Stein's method yield a Berry-Esseen theorem for the number of urns having occupancy $d \geq 2$ when $n$ balls are uniformly distributed over $m$ urns. In particular, there exists a constant $C$ depending only on $d$ such that$$\sup_{z \in \mathbb{R}}\left|P\left( W_{n,m} \le z \right) -P(Z \le z)\right| \le C \frac{\sigma_{n,m}}{1+(\frac{n}{m})^3} \quad\mbox{for all $n \ge d$ and $m \ge 2$,}$$where $W_{n,m}$ and $\sigma_{n,m}^2$ are the standardized count and variance, respectively, of the number of urns with $d$ balls, and $Z$ is a standard normal random variable. Asymptotically, the bound is optimal up to constants if $n$ and $m$ tend to infinity together in a way such that $n/m$ stays bounded.