A Berry - Esseen bound for the uniform multinomial occupancy model ∗

The inductive size bias coupling technique and Stein’s method yield a Berry-Esseen theorem for the number of urns having occupancy d ≥ 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∈R |P (Wn,m ≤ z)− P (Z ≤ z)| ≤ C σn,m 1 + ( n m )3 for all n ≥ d and m ≥ 2, where Wn,m and σ 2 n,m 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.