The number of k-tons in the coupon collector problem

TL;DR

This paper analyzes the asymptotic distribution of the number of k-tons, coupons seen k times by the time all coupon types are collected m times, in a generalized coupon collector problem.

Contribution

It determines the asymptotic distribution of k-tons and their joint distribution with total coupons collected for fixed m and various k values.

Findings

Derived the asymptotic distribution of k-tons for fixed m.

Established joint probability distribution of k-tons and total coupons.

Extended understanding of coupon collector dynamics in probabilistic terms.

Abstract

Consider the coupon collector problem where each box of a brand of cereal contains a coupon and there are n different types of coupons. Suppose that the probability of a box containing a coupon of a specific type is $1/n$ and that we keep buying boxes until we collect at least $m$ coupons of each type. For $k\geq m$ call a certain coupon a $k$-ton if we see it $k$ times by the time we have seen $m$ copies of all of the coupons. Here we determine the asymptotic distribution of the number of $k$-tons after we have collected $m$ copies of each coupon for any $k$ in a restricted range, given any fixed $m$. We also determine the asymptotic joint probability distribution over such values of $k$ and the total number of coupons collected.