Convergence of point processes associated with coupon collector's and Dixie cup problems

We prove that, in the coupon collector’s problem, the point processes given by the times of $r$-th arrivals for coupons of each type, centered and normalized in a proper way, converge toward a non-homogeneous Poisson point process. This result is then used to derive some generalizations and infinite-dimensional extensions of classical limit theorems on the topic.