The Multinomial Allocation Model and the Size of a Randomly Chosen Box

Abstract: We revisit the random allocation of $n$ balls into $N$ boxes with probabilities $q_1,\ldots,q_N$, focusing on the proportion $\hat q_r$ of boxes containing exactly $r$ balls. Classical asymptotic results for the expectations, variances, and covariances of these proportions are reformulated in terms of the size distribution of a randomly chosen box. We further derive explicit two-sided bounds for the associated remainder terms, allowing for weaker assumptions than those previously required.