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

Limit distributions for the problem of collecting pairs