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2013 A Berry-Esseen bound for the uniform multinomial occupancy model
Jay Bartroff, Larry Goldstein
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Electron. J. Probab. 18: 1-29 (2013). DOI: 10.1214/EJP.v18-1983


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.


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Jay Bartroff. Larry Goldstein. "A Berry-Esseen bound for the uniform multinomial occupancy model." Electron. J. Probab. 18 1 - 29, 2013.


Accepted: 17 February 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1287.60031
MathSciNet: MR3035755
Digital Object Identifier: 10.1214/EJP.v18-1983

Primary: 60F05
Secondary: 60C05

Keywords: coupling , size bias , Stein’s method , urn models

Vol.18 • 2013
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