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.Comment: Typo corrected in abstrac