Normal approximation for isolated balls in an urn allocation model

الأصل · EN

Consider throwing n balls at random into m urns, each ball landing in urn i with probability pᵢ. 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 (pᵢ, 1 ≤ i ≤ m) vary in such a way that ᵢ pᵢ = O(n⁻¹), then S satisfies a CLT if and only if n² ∑ᵢ pᵢ² tends to infinity, and demonstrate an optimal rate of convergence in the CLT in this case. In the uniform case (pᵢ ≡ 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.