Poisson limit theorems in allocation schemes of distinguishable particles

Abstract

We consider a random variable $\mu_r(n, K, N)$ being the number of cells containing $r$ particles among first $K$ cells in an equiprobable allocation scheme of at most $n$ distinguishable particles over $N$ different cells. We find conditions ensuring the convergence of these random variables to a random Poisson variable. We describe a limit distribution. These conditions are of a simplest form, when the number of particles $r$ belongs to a bounded set or as $K$ is equivalent to $\sqrt{N}$. Then random variables $\mu_r(n, K, N)$ behave as the sums of independent identically distributed indicators, namely, as binomial random variables, and our conditions coincide with the conditions of a classical Poisson limit theorem. We obtain analogues of these theorems for an equiprobable allocation scheme of $n$ distinguishable particles of $N$ different cells. The proofs of these theorems are based on the Poisson limit theorem for the sums of exchangeable indicators and on an analogue of the local limit Gnedenko theorem.