@article{EJP699,
author = {Mathew Penrose},
title = {Normal Approximation for Isolated Balls in an Urn Allocation Model},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {14},
year = {2009},
keywords = {Berry-Esseen bound, central limit theorem, occupancy scheme, size biased coupling, Stein's method.},
abstract = {Consider throwing $n$ balls at random into $m$ urns, each ball landing in urn $i$ with probability $p(i)$. Let $S$ be the resulting number of singletons, i.e., urns containing just one ball. We give an error bound for the Kolmogorov distance from the distribution of $S$ to the normal, and estimates on its variance. These show that if $n$, $m$ and $(p(i))$ vary in such a way that $n p(i)$ remains bounded uniformly in $n$ and $i$, then $S$ satisfies a CLT if and only if ($n$ squared) times the sum of the squares of the entries $p(i)$ tends to infinity, and demonstrate an optimal rate of convergence in the CLT in this case. In the uniform case with all $p(i)$ equal and with $m$ and $n$ growing proportionately, we provide bounds with better asymptotic constants. The proof of the error bounds is based on Stein's method via size-biased couplings.},
pages = {no. 74, 2155-2181},
issn = {1083-6489},
doi = {10.1214/EJP.v14-699},
url = {http://ejp.ejpecp.org/article/view/699}}