A Limit Theorem for Random Allocations

A limit theorem is presented for random allocations. For a fixed period we allocate $m$ balls into $N$ boxes. We repeat the experiment throughout $n$ periods. Let $p_q$ denote the probability that we do not place more than $q$ balls into any of the $N$ boxes during any of the $n$ repetitions. The limit of $p_q$ is determined when $m,n,N o infty$.