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Normal approximation for isolated balls in an urn allocation model. (English) Zbl 1191.60029

Summary: 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.

MSC:

60F05 Central limit and other weak theorems
60C05 Combinatorial probability
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