A Berry-Esseen bound for the uniform multinomial occupancy model

TL;DR

This paper establishes a Berry-Esseen bound for the distribution of the number of urns with a fixed occupancy in a uniform multinomial model, using Stein's method and size bias coupling.

Contribution

It provides a new Berry-Esseen theorem with explicit bounds for the occupancy count distribution in the multinomial model, improving understanding of its normal approximation.

Findings

The bound is optimal up to constants when n and m grow with n/m bounded.

The theorem applies for all n ≥ d and m ≥ 2, with a constant depending only on d.

Explicit rate of convergence in the normal approximation is established.

Abstract

The inductive size bias coupling technique and Stein's method yield a Berry-Esseen theorem for the number of urns having occupancy $d \ge 2$ when $n$ balls are uniformly distributed over $m$ urns. In particular, there exists a constant $C$ depending only on $d$ such that $$ \sup_{z \in \mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left( \frac{1+(\frac{n}{m})^3}{\sigma_{n,m}} \right) \quad \mbox{for all $n \ge d$ and $m \ge 2$,} $$ 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.