  The following variant of the collector's problem has attracted considerable
attention relatively recently (see, e.g., N. Pintacuda 1980, D. Foata H.
Guo-Niu and B. Lass 2001, D. Foata and D. Zeilberger 2003, I. Adler, S. Oren
and S. Ross 2003, and S. Ross 2010): There is one main collector who collects
coupons. Assume there are $N$ different types of coupons with, in general,
unequal occurring probabilities. When the main collector gets a "double", she
gives it to her older brother; when this brother gets a "double", he gives it
to the next brother, and so on. Hence, when the main collector completes her
collection, the album of the $j$-th sibling, $j = 2, 3, \dots$, will still have
$U_j^N$ empty spaces. In this article we develop techniques of computing
asymptotics of the average $E[U_j^N]$ of $U_j^N$ as $N \rightarrow \infty,$ for
a large class of families of coupon probabilities. We also give various
illustrative examples.
