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 pi. 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 (pi, 1 ≤ i ≤ m) vary in such a way that supi pi = O(n− 1), then S satisfies a CLT if and only if n2 Σ i p2i tends to infinity, and demonstrate an optimal rate of convergence in the CLT in this case. In the uniform case (pi ≡ m− 1) 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 coupling. © 2009 Applied Probability Trust.