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Normal Approximation for Isolated Balls in an Urn Allocation Model

2009
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Electronic Journal of Probability
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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

doi:10.1214/ejp.v14-699
fatcat:jxcnhayofbb6rcge6zt5oarelu