  We consider an extended variant of the classical coupon collector's problem
with infinite number of collections. An arriving coupon is placed in the
$r^{th}$ collection, $r\ge0$, if $r$ is the smallest index such that the
corresponding collection still does not have a coupon of this type. We derive
distributional limit theorems for the number of empty spots in different
collections at the time when the $0^{th}$ collection was completed, as well as
after some delay. We also obtain limiting distributions for completion times of
different collections. All main results are given in an ultimate
infinite-dimensional form in the sense of distributional convergence in
$\mathbb R^\infty$. The main tool in the proofs is convergence of specially
constructed point processes.
