Extremality and Limit Laws for the Siblings of the Coupon Collector

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 $j\ge2$, $\E U_j^N$ 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 $2,\ldots,J$, the naturally normalized vector converges jointly to $(W,\ldots,W)$, 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.