Normal approximation for isolated balls in an urn allocation model

Details Normal approximation for isolated balls in an urn allocation model Normal approximation for isolated balls in an urn allocation model

2009-01-22

arxiv.org/abs/0901.3493v1

Mathematics

Probability

32 Pages

Scientific paper

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 $S$ to the normal, and estimates on its variance. These show that if $n$, $m$ and $(p_i, 1 \leq i \leq m)$ vary in such a way that $\sup_i p_i = O(n^{-1})$, then $S$ satisfies a CLT if and only if $n^2 \sum_i p_i^2$ tends to infinity, and demonstrate an optimal rate of convergence in the CLT in this case. In the uniform case $(p_i \equiv m^{-1}) with $m$ and $n$ growing proportionately, we provide bounds with better asymptotic constants. The proof of the error bounds are based on Stein's method via size-biased couplings.

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