Normal Approximation for Isolated Balls in an Urn Allocation Model

Mathew Penrose
2009 Electronic Journal of Probability  
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 , 1 ≤ i ≤ m) vary in such a way that sup i p i = O(n −1 ), then S satisfies a CLT if and only if n 2 i p 2 i tends to infinity, and demonstrate an optimal rate
more » ... convergence in the CLT in this case. In the uniform case (p i ≡ 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.
doi:10.1214/ejp.v14-699 fatcat:jxcnhayofbb6rcge6zt5oarelu