On some distributions connected with the waiting time in a polynomial scheme

Let, in a polynomial scheme with $N$ equiprobable outcomes, $n$ trials be made, and $\rho_1(n)$ $(\rho_2(n))$ denote the maximum (minimum) sampling frequencies. We consider $(\rho_1(n),\rho_2(n))$ as a random function of time parameter $n$ and study the asymptotic behaviour (as $N\to\infty$) of the random variables $\tau_m=\nu_2(m)-\nu_1(m)$, $\rho_1(\nu_2(m))$ and $\rho_2(\nu_1(m))$, where
$$ \nu_i(m)=\min\{n\colon\rho_i(n)=m\};\quad i=1,2;\quad m\ge1. $$