Dixie cup problem in an interlacing process

TL;DR

This paper analyzes a generalized Dixie cup problem with a mixed coupon distribution, deriving asymptotic expectations and moments, and extends the problem to multiple subfamilies.

Contribution

It introduces a new analysis for the Dixie cup problem with mixed distributions and provides asymptotic results for large coupon sets.

Findings

Asymptotic expectation derived for mixed distributions

Both distributions influence the asymptotics

Generalization to multiple subfamilies

Abstract

The double Dixie cup problem of D.J. Newman and L. Shepp is a well-known variant of the coupon collector problem, where the object of study is the number of coupons that a collector has to buy in order to complete m sets of all N existing different coupons. In this paper we consider the case where the coupons distribution is a mixture of two different distributions, where the coupons from the first distribution are far rarer than the ones coming from the second. We apply a Poissonization technique, as well as well known results and techniques from our previous work, to derive the asymptotics (leading term) of the expectation of the above random variable as N goes to infinity for large classes of distributions. As it turns out, both distributions contribute to this result. The leading asymptotics of the rising moments of the aforementioned random variable are also discussed. We conclude by generalizing the problem to the case where the family of coupons is a mixture of j subfamilies.