The multinomial allocation model provides a natural framework for a generalized birthday problem, in which n balls are distributed among N boxes with non-uniform allocation probabilities. As a classical object in probability theory, it also appears in a wide range of applications.

In this talk, I revisit an asymptotic result due to Kolchin, Sevastyanov, and Chistyakov from the 1970s and reformulate it in terms of the size of a randomly selected box. This viewpoint leads to a strengthened version of the result, where explicit two-sided bounds on the remainder terms can be obtained.