Normal approximation for isolated balls in an urn allocation model

TL;DR

This paper analyzes the distribution of the number of singletons in an urn model, providing error bounds for normal approximation and conditions for a CLT, using Stein's method.

Contribution

It offers new error bounds and variance estimates for the normal approximation of singleton counts in urn models, with conditions for CLT validity.

Findings

Error bounds for the Kolmogorov distance to normal distribution.

Conditions under which the singleton count satisfies a CLT.

Optimal convergence rates in the CLT for the model.

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 $S$ 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.