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<p>The “double Dixie cup problem” of D.J. Newman and L. Shepp is a well-known variant of the coupon collector’s problem, where the object of study is the number of coupons that a collector has to buy in order to complete <i>m</i> sets of all <i>N</i> existing different coupons. In this paper we consider the case where the coupons distribution is a <i>mixture</i> 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N\rightarrow \infty\)</EquationSource> </InlineEquation> 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 <i>j</i> subfamilies.</p>

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Dixie Cup Problem in an Interlacing Process

  • 期刊论文

  • 发表日期：

    2025年11月18日

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  • Aristides V. Doumas

摘要

The “double Dixie cup problem” of D.J. Newman and L. Shepp is a well-known variant of the coupon collector’s 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 \(N\rightarrow \infty\) 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.