Limit distributions for the problem of collecting pairs

Let $N_n=\{1,2,...,n\}$. Elements are drawn from the set $N_n$ with replacement, assuming that each element has probability $1/n$ of being drawn. We determine the limiting distributions for the waiting time until the given portion of pairs $jj$, $j\in N_n$, is sampled. Exact distributions of some related random variables and their characteristics are also obtained.