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

TL;DR

This paper analyzes the asymptotic expected number of draws needed in a group-drawing variant of the Coupon Collector's Problem under uniform distribution, covering different regimes of group size.

Contribution

It provides precise asymptotic expressions for the expected collection time across three regimes of group size in the uniform coupon collection problem.

Findings

Derived asymptotic formulas for constant group size s.

Extended analysis to s proportional to n.

Analyzed the case where s is close to n.

Abstract

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