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

In a generalized allocation scheme of $n$ particles over $N$ cells we consider the random variable $\eta_{n,N}(K)$ which is the number of particles in a given set consisting of $K$ cells. We prove that if $n, K, N\to\infty$, then under some conditions random variables $\eta_{n,N}(K)$ are asymptotically normal, and under another conditions $\eta_{n,N}(K)$ converge in distribution to a Poisson random variable. For the case when $N\to\infty$ and $n$ is a fixed number, we find conditions under which $\eta_{n,N}(K)$ converge in distribution to a binomial random variable with parameters $n$ and $s=\frac{K}{N}$, $0<K<N$, 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 $B(\beta)=\ln(1-\beta)$ the conditions $n,N,K\to\infty$, $\frac{K}{N}\to s$, $N=\gamma\ln(n)+o(\ln(n))$, where $0< s<1$, $0<\gamma<\infty$, are satisfied, then distributions of random variables $\frac{\eta_{n,N}(K)}{n}$ converge to a beta-distribution with parameters $s\gamma$ and $(1-s)\gamma$.