论文标题
通过Stein的方法通过小组图处理优惠券收集器的问题
Approaching the coupon collector's problem with group drawings via Stein's method
论文作者
论文摘要
在本文中，研究了优惠券收集器在组图中的问题。假设有$ n $不同的优惠券。每次绘制$ n $优惠券的$ s $时，所有选择都应该具有同等的概率。重点在于波动，如$ n \ to \ infty $，数量$ z_ {n，s}（k_n）$的优惠券，这些优惠券在第一个$ k_n $图纸中尚未绘制。使用$ z_ {n，s}（n，s}（k_n）$的定量中心限制定理，使用尺寸偏置的耦合结构以及Stein的方法，显示了$ k_n = {n \ over s}（α\ log log（n）+x）$，其中$ 0 <α<α<α<1 $和$ x \ bb，同样的耦合结构用于在边界情况下再次使用Stein的方法检索定量泊松限值定理。
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 $k_n={n\over s}(α\log(n)+x)$, where $0<α<1$ and $x\in\mathbb{R}$. The same coupling construction is used to retrieve a quantitative Poisson limit theorem in the boundary case $α=1$, again using Stein's method.
