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

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.

### MSC:

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

### Keywords:

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