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Extreme Value Distributions for Random Coupon Collector and Birthday Problems

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Abstract

Take n independent copies of a strictly positive random variable X and divide each copy with the sum of the copies, thus obtaining n random probabilities summing to one. These probabilities are used in independent multinomial trials with n outcomes. Let N n(N * n) be the number of trials needed until each (some) outcome has occurred at least c times. By embedding the sampling procedure in a Poisson point process the distributions of N n and N * n can be expressed using extremes of independent identically distributed random variables. Using this, asymptotic distributions as n → ∞ are obtained from classical extreme value theory. The limits are determined by the behavior of the Laplace transform of X close to the origin or at infinity. Some examples are studied in detail.

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Authors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology, SE–10044, Stockholm, Sweden

    Lars Holst

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  1. Lars Holst
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Holst, L. Extreme Value Distributions for Random Coupon Collector and Birthday Problems. Extremes 4, 129–145 (2001). https://doi.org/10.1023/A:1013921125928

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  • DOI: https://doi.org/10.1023/A:1013921125928

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