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

Abstract

AbstractWe study the coupon collector’s problem with group drawings. 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 $k_n=({n/s})(\alpha\log(n)+x)$ , 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.