Normal Approximation for Isolated Balls in an Urn Allocation Model

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.