Asymptotic behavior of maximum of an equiprobable polynomial scheme

An equiprobable polynomial scheme is considered with $N$ outcomes and $n$ independent trials. For $n/N\ln N\to\infty$ the asymptotic distribution is studied of the quantity $\rho=\max\limits_{1\leqslant i\leqslant N}\xi_i$, где $\xi_i$, where $\xi_i$ is the number of occurrences of the $i$-th outcome.