Оn the distribution of the time up to the first occurence of a given number of different $l$-tuple series

Let $\nu_l(N,k)$ ($\nu_{\ge l}(N,k)$) be the number of trials up to the first occurence of a series of $l$ (not less than $l$) outcomes containing given $k$ from $N$ possible outcomes. Let $\xi_l(N,n)$ ($\xi_{\ge l}(N,n)$) be the number of outcomes for which no $l$-tuple series (no series of length equal to or greater than $l$) occurred in $n$ trials.
Asymptotic behaviour, as $N\to\infty$, of $\nu_l(N,k)$, $\nu_{\ge l}(N,k)$, $\xi_l(N,n)$, and $\xi_{\ge l}(N,n)$ is studied for various relations between $k$, $n$ and $N$.