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.

Key words and phrases. Poisson distribution, power series distribution, limit theorem, generalized scheme of allocations.