Extremality and Limit Laws for the Siblings of the Coupon Collector

Abstract

We study the siblings version of the coupon collector problem. A main collector stops when every coupon type has appeared at least once, duplicates are passed successively to later siblings, and $U_j^N$ denotes the number of empty spaces in collector $j$'s album at the main completion time. We prove three results. First, for every fixed $N$ and $j\ge 2$, $\E U_j^N$ is uniquely maximized over positive coupon distributions by the uniform distribution; in fact it decreases strictly along every nonconstant ray from the uniform vector. Second, in the uniform model, $U_j^N$ is stochastically increasing in $N$, and we construct an increasing coupling using top spacings of exponential order statistics. Third, for fixed album indices $2,\dots ,J$, the naturally normalized vector converges jointly to $(W,\dots ,W)$, where $W$ is exponential with mean one. We also derive exact Poissonized and alternating-subset formulae and give a transfer principle for leading expectation asymptotics.

24 pages, 0 figures. Submitted for publication prior to the appearance of the Doumas-Spektor preprint arXiv:2606.21591. This version adds a citation and comparison note explaining the overlap in the finite-N radial expectation-extremality theorem