论文标题
限制扩展优惠券收集器问题中的定理
Limit theorems in the extended coupon collector's problem
论文作者
论文摘要
我们考虑了经典优惠券收藏家与无限收藏数量的问题的扩展变体。如果$ r $是最小的索引，则将到达的优惠券放置在$ r^{th} $集合中，$ r \ ge0 $，因此相应的集合仍然没有这种类型的优惠券。我们在$ 0^{th} $集合完成时以及一定的延迟之后得出不同集合中空位数的分布限制定理。我们还获得了限制分布，以完成不同收藏的完成时间。所有主要结果均以$ \ mathbb r^\ infty $中的分布收敛意义以最终的无限尺寸形式给出。证明的主要工具是专门构建的点过程的收敛性。
We consider an extended variant of the classical coupon collector's problem with infinite number of collections. An arriving coupon is placed in the $r^{th}$ collection, $r\ge0$, if $r$ is the smallest index such that the corresponding collection still does not have a coupon of this type. We derive distributional limit theorems for the number of empty spots in different collections at the time when the $0^{th}$ collection was completed, as well as after some delay. We also obtain limiting distributions for completion times of different collections. All main results are given in an ultimate infinite-dimensional form in the sense of distributional convergence in $\mathbb R^\infty$. The main tool in the proofs is convergence of specially constructed point processes.
