Approaching the coupon collector's problem with group drawings via Stein's method

TL;DR

This paper analyzes the fluctuations in the coupon collector's problem with group drawings, providing quantitative limit theorems using Stein's method for large n, including normal and Poisson approximations.

Contribution

It introduces a size-biased coupling construction combined with Stein's method to derive precise limit theorems for the coupon collector's problem with group drawings.

Findings

Established a normal approximation for the number of uncollected coupons when lpha<1.

Derived a Poisson limit theorem in the boundary case lpha=1.

Provided quantitative bounds for the convergence in both cases.

Abstract

In this paper the coupon collector's problem with group drawings is studied. Assume there are $ n $ different coupons. At each time precisely $ s $ of the $ n $ coupons are drawn, where all choices are supposed to have equal probability. The focus lies on the fluctuations, as $n\to\infty$, of the number $Z_{n,s}(k_n)$ of coupons that have not been drawn in the first $k_n$ drawings. Using a size-biased coupling construction together with Stein's method for normal approximation, a quantitative central limit theorem for $Z_{n,s}(k_n)$ is shown for the case that $k_n={n\over s}(\alpha\log(n)+x)$, where $0<\alpha<1$ and $x\in\mathbb{R}$. The same coupling construction is used to retrieve a quantitative Poisson limit theorem in the boundary case $\alpha=1$, again using Stein's method.