The Multinomial Allocation Model and the Random Box Load

TL;DR

This paper revisits the classical random allocation model, providing a clearer reformulation of asymptotic results and deriving explicit bounds for remainder terms under weaker assumptions.

Contribution

It offers a new, transparent formulation of asymptotic expectations and variances, along with explicit bounds, improving upon previous results in the random allocation model.

Findings

Reformulated classical asymptotic results in a compact form.

Derived explicit two-sided bounds for remainder terms.

Weaker assumptions enable broader applicability of bounds.

Abstract

We revisit the random allocation model in which $n$ balls are independently placed into $N$ boxes with probabilities $q_1,\ldots,q_N$. A classical asymptotic result due to Kolchin, Sevastyanov, and Chistyakov for the expectations, variances, and covariances of the occupancy counts is reformulated in a compact and transparent form in terms of the load of a randomly selected box. We further derive explicit two-sided bounds for the associated remainder terms, obtained under weaker assumptions than those previously required.