Radial Transform Extremality for the Siblings of the Coupon Collector

Abstract

In the siblings version of the coupon collector, a main collector stops when every coupon type has appeared once. Duplicates are passed successively to siblings, and $U_j^N$ denotes the number of empty spaces in the $j$th collector's album at the main completion time. We prove finite-$N$ radial transform strengthenings of the uniform-probability extremality principle. For every $N\ge2$, every $j\ge2$, every positive nonuniform probability vector $p$, and the ray $p(\theta)=u+\theta(p-u)$ from the uniform vector $u$, the full probability generating function $\mathbb{E}_{p(\theta)}z^{U_j^N}$ is strictly decreasing in $\theta$ for $z>1$ and strictly increasing in $\theta$ for $0<z<1$. Thus the same full PGF has opposite radial monotonicity on the two sides of $z=1$, the left side giving a radial Laplace-transform order. At the coefficient level, along every nonconstant ray from the uniform vector, uniform probabilities maximize every binomial moment of $U_j^N$, equivalently giving a finite absolutely-monotone/binomial-transform order. The proof of the right-PGF and binomial-moment theorem is exact and finite-dimensional. It uses Poissonization, a marked Poissonized PGF identity, a normalized alternating subset expansion, and a positive-kernel radial derivative formula obtained from a local cumulative-polynomial dissipation lemma. The Laplace-transform theorem follows from a separate Gamma-mixture race representation.