Tomer Sher | Searcharxiv

arXiv2026

Asymptotic Results for Uniform Group Drawing in the Coupon Collector's Problem

The article explores the asymptotic behavior of the expected number of drawings in the Coupon Collector's Problem with group-drawing under the uniform distribution. In this variant, each draw consists of a package of $s$ distinct coupons selected uniformly at random from a set of $n$ coupons. We focus on three regimes of the package size $s$: (i) constant $s$, (ii) $s$ proportional to $n$, and (iii) $s$ "very close" to $n$. For each case, we provide precise asymptotic expressions for the expected collection time. Keywords: Coupon Collector's Problem, Group Drawings, Uniform Distribution, Asymptotic Analysis, Expected Collection Time

math.PR↗

arXiv2025

On a Conjecture on Uniform Group Drawings in the Coupon Collector Problem

We address a conjecture of Schilling concerning the optimality of the uniform distribution in the generalized Coupon Collector's Problem (CCP) where, in each round, a subset (package) of $s$ coupons is drawn from a total of $n$ distinct coupons. While the classical CCP (with single-coupon draws) is well understood, the group-draw variant, where packages of size $s$ are drawn, presents new challenges and has applications in areas such as biological network models. Consider the set of all distributions over the collection of $\binom{n}{s}$ packages of size $s$. Schilling showed that, for $s=n-1$, the uniform distribution yields the minimal expected time for collecting all coupons. She further conjectured that, for $2\le s\le n-2$, the uniform distribution does not yield the minimum. We prove Schilling's conjecture in full by presenting "natural" non-uniform distributions yielding strictly lower expected collection times. Explicit formulas are provided for the expected number of rounds under these and related distributions Keywords: Coupon Collector's Problem, Group Drawings, Uniform Distribution, Expected Collection Time, Schilling's Conjecture, Optimal Distribution.

math.PR↗