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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 74 open journal systems 


Normal approximation for isolated balls in an urn allocation model

Mathew D. Penrose, University of Bath


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.


Full text: PDF

Pages: 2155-2181

Published on: October 6, 2009
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Electronic Journal of Probability. ISSN: 1083-6489