POISSON LIMIT THEOREMS IN ALLOCATION SCHEMES OF DISTINGUISHABLE PARTICLES

Abstract

© 2020 Abdushukurov F.A. We consider a random variable µr(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 $$. Then random variables µr(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.