On the limiting behaviour of extreme order statistics in polynomial scheme

A sequence of $n$ independent trials with $N$ equiprobable events $E_1,\dots,E_N$ is considered. Let $\eta_i$ be the number of occurances of the event $E_i$ and $\eta_{(i)}$ be the $i$th order statistic for the sample $\eta_1,\dots,\eta_N$. The asymptotic behaviour of $\eta_{(i)}$ and $\eta_{(N-i+1)}$ as $n,\ N\to\infty$ is investigated by means of conversion to the analogous problem for order statistics for independent Poisson random variables.