A Berry-Esseen bound for the uniform multinomial occupancy model. (English) Zbl 1287.60031

Bartroff, Jay; Goldstein, Larry

Electron. J. Probab. 18, Paper No. 27, 29 p. (2013).

Authors’ abstract: The inductive size bias coupling technique and Stein’s method yield a Berry-Esseen theorem for the number of urns having occupancy \(d\geq2\), when \(n\) balls are uniformly distributed over \(m\) urns. In particular, there exists a constant \(C\) depending only on \(d\) such that \[ \sup_{x\in\mathbb{R}}\left| \operatorname P\left( W_{n,m}\leq z\right) - \operatorname P\left( Z\leq z\right) \right| \leq C\frac{\sigma_{n,m}}{1+\left( \frac{n}{m}\right) ^{3}} \] for all \(n\geq d\) and \(m\geq2\), where \(W_{n,m}\) and \(\sigma_{n,m}^{2}\) are 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.

Reviewer: Wiesław Dziubdziela (Miedziana Gora)

MSC:

60F05	Central limit and other weak theorems
60C05	Combinatorial probability

Keywords:

Stein’s method; size bias; coupling; urn models