  The "double Dixie cup problem" of D.J. Newman and L. Shepp (1960) is a
well-known variant of the coupon collector's problem, where the object of study
is the number $T_{m}(N)$ of coupons that a collector has to buy in order to
complete $m$ sets of all $N$ existing different coupons. More precisely, the
problem is to determine the asymptotics of the expectation (and the variance)
of $T_{m}(N)$, as well as its limit distribution, as the number $N$ of
different coupons becomes arbitrarily large. The classical case of the problem,
namely the case of equal coupon probabilities, is here extended to the general
case, where the probabilities of the selected coupons are unequal.
  In the beginning of the article we give a brief review of the formulas for
the moments and the moment generating function of the random variable
$T_{m}(N)$. Then, we develop techniques of computing the asymptotics of the
first and the second moment of $T_{m}(N)$ (our techniques apply to the higher
moments of $T_{m}(N)$ as well). From these asymptotic formulas we obtain the
leading behavior of the variance $V[\,T_{m}(N)\,]$ as $N \to \infty$. Finally,
based on the asymptotics of $E[\,T_{m}(N)\,]$ and $V[\,T_{m}(N)\,]$ we obtain
the limit distribution of the random variable $T_{m}(N)$ for large classes of
coupon probabilities. As it turns out, in many cases, albeit not always,
$T_{m}(N)$ (appropriately normalized) converges in distribution to a Gumbel
random variable. Our results on the limit distribution of $T_{m}(N)$ generalize
a well-known result of P. Erd\H{o}s and A. R\'{e}nyi (1961) regarding the limit
distribution of $T_{m}(N)$ for the case of equal coupon probabilities.
