The Coupon Collector's Problem Revisited: Generalizing the Double Dixie Cup Problem of Newman and Shepp

TL;DR

This paper extends the classical coupon collector's problem to cases with unequal coupon probabilities, analyzing the asymptotics of the number of coupons needed to collect multiple sets and identifying the limit distribution as often Gumbel.

Contribution

It generalizes the known results for equal probabilities to unequal probabilities, providing asymptotic formulas and limit distributions for the generalized problem.

Findings

Asymptotic formulas for the expectation and variance of T_m(N)

Limit distribution of T_m(N) often converges to Gumbel

Techniques applicable to higher moments of T_m(N)

Abstract

The "double Dixie cup problem" of D.J. Newman and L. Shepp (1960) is a well-known variant of the coupon collector's problem, where the object of study is the number $T_{m}(N)$ of coupons that a collector has to buy in order to complete $m$ sets of all $N$ existing different coupons. More precisely, the problem is to determine the asymptotics of the expectation (and the variance) of $T_{m}(N)$, as well as its limit distribution, as the number $N$ of different coupons becomes arbitrarily large. The classical case of the problem, namely the case of equal coupon probabilities, is here extended to the general case, where the probabilities of the selected coupons are unequal. In the beginning of the article we give a brief review of the formulas for the moments and the moment generating function of the random variable $T_{m}(N)$. Then, we develop techniques of computing the asymptotics of the first and the second moment of $T_{m}(N)$ (our techniques apply to the higher moments of $T_{m}(N)$ as well). From these asymptotic formulas we obtain the leading behavior of the variance $V[\,T_{m}(N)\,]$ as $N \to \infty$. Finally, based on the asymptotics of $E[\,T_{m}(N)\,]$ and $V[\,T_{m}(N)\,]$ we obtain the limit distribution of the random variable $T_{m}(N)$ for large classes of coupon probabilities. As it turns out, in many cases, albeit not always, $T_{m}(N)$ (appropriately normalized) converges in distribution to a Gumbel random variable. Our results on the limit distribution of $T_{m}(N)$ generalize a well-known result of P. Erd\H{o}s and A. R\'{e}nyi (1961) regarding the limit distribution of $T_{m}(N)$ for the case of equal coupon probabilities.