On the number of particles from a marked set of cells for an analogue of a general allocation scheme

In a general scheme of allocation of no more than $n$ particles to $N$ cells we prove limit theorems for the random variable $\eta_{n,N}(K)$ which is the number of particles in a given set of $K$ cells. The main result of the paper is Theorem 1. Limit distribution in this theorem depends on $s=\lim\frac{K}{N}$. If $0<s<1$, then the limit distribution is that of the minimum of independent Gaussian random variables, and if $s=1$, then it is the distribution of the absolute value of a Gaussian random variable taken with the minus sign.