The Multinomial Allocation Model and the Random Box Load

We revisit the random allocation model in which $n$ balls are independently placed into $N$ boxes with probabilities $q_1,\dots ,q_N$. 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.