Poisson limit theorems for the generalized allocation scheme

Abstract. Poisson limit theorems are studied for discrete probability models. Consider the allocation of \(n\) balls into \(N\) boxes. Let \(\mu_r(N,K,n)\) denote the number of those boxes from the first \(K\) boxes which contain \(r\) balls. Convergence of \(\mu_r(N,K,n)\) to a Poisson distribution is proved as \(K, n\to\infty\). Analogous results are obtained for the generalized allocation scheme and also for several discrete probabilistic models.