Electronic Journal of Probability

A Berry-Esseen bound for the uniform multinomial occupancy model

Jay Bartroff and Larry Goldstein

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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.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 27, 29 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

Stein’s method size bias coupling urn models

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Bartroff, Jay; Goldstein, Larry. A Berry-Esseen bound for the uniform multinomial occupancy model. Electron. J. Probab. 18 (2013), paper no. 27, 29 pp. doi:10.1214/EJP.v18-1983. https://projecteuclid.org/euclid.ejp/1465064252

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