On a number of particles in a marked set of cells in a general allocation scheme

Abstract In a generalized allocation scheme of n particles over N cells we consider the random variable ηn,N(K) which is the number of particles in a given set consisting of K cells. We prove that if n, K, N → ∞, then under some conditions random variables ηn,N(K) are asymptotically normal, and under another conditions ηn,N(K) converge in distribution to a Poisson random variable. For the case when N → ∞ and n is a fixed number, we find conditions under which ηn,N(K) converge in distribution to a binomial random variable with parameters n and s = KN $\begin{array}{} \displaystyle \frac{K}{N} \end{array}$, 0 < K < N, multiplied by a integer coefficient. It is shown that if for a generalized allocation scheme of n particles over N cells with random variables having a power series distribution defined by the function B(β) = ln(1 − β) the conditions n, N, K → ∞, KN $\begin{array}{} \displaystyle \frac{K}{N} \end{array}$ → s, N = γ ln(n) + o(ln(n)), where 0 < s < 1, 0 < γ < ∞, are satisfied, then distributions of random variables ηn,N(K)n $\begin{array}{} \displaystyle \frac{\eta_{n,N}(K)}{n} \end{array}$ converge to a beta-distribution with parameters sγ and (1 − s)γ.