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 $d\ge 2$ when $n$ balls are uniformly distributed over $m$ urns. In particular, there exists a constant $C$ depending only on $d$ such that $$ \sup_{z \in \mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left( \frac{1+(\frac{n}{m})^3}{σ_{n,m}} \right) \quad \mbox{for all $n \ge d$ and $m \ge 2$,} $$ 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.

Typo corrected in abstract

References in corpus (4)

• Normal approximation under local dependence
• Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws
• Local limit theorems for finite and infinite urn models
• Stein couplings for normal approximation

Cited by in corpus (3)

• Discretized normal approximation by Stein's method
• Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process
• Pólya urns with immigration at random times