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New Results on a Generalized Coupon Collector Problem Using Markov Chains

  • Emmanuelle Anceaume (a1), Yann Busnel (a2) and Bruno Sericola (a3)

Abstract

In this paper we study a generalized coupon collector problem, which consists of determining the distribution and the moments of the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we obtain expressions for the distribution and the moments of this time. We also prove that the almost-uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which minimizes the expected time to obtain a fixed subset of distinct coupons. This optimization result is extended to the complementary distribution of the time needed to obtain the full collection, proving by the way this well-known conjecture. Finally, we propose a new conjecture which expresses the fact that the almost-uniform distribution should minimize the complementary distribution of the time needed to obtain any fixed number of distinct coupons.

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Copyright

Corresponding author

Postal address: CNRS, Campus de Beaulieu, 35042 Rennes Cedex, France.
∗∗ Postal address: Université de Nantes, 2 rue de la Houssinière, 44322 Nantes Cedex 03, France.
∗∗∗ Postal address: INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France. Email address: bruno.sericola@inria.fr

References

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[1] Boneh, A. and Hofri, M. (1997). The coupon-collector problem revisited—a survey of engineering problems and computational methods. Commun. Statist. Stoch. Models 13, 3966.
[2] Brown, M., Peköz, E. A. and Ross, S. M. (2008). Coupon collecting. Prob. Eng. Inf. Sci. 22, 221229.
[3] Doumas, A. V. and Papanicolaou, V. G. (2012). The coupon collector's problem revisited: asymptotics of the variance. Adv. Appl. Prob. 44, 166195.
[4] Doumas, A. V. and Papanicolaou, V. G. (2013). Asymptotics of the rising moments for the coupon collector's problem. Electron. J. Prob. 18, 15pp.
[5] Flajolet, P., Gardy, D. and Thimonier, L. (1992). Birthday paradox, coupon collectors, caching algorithms and self-organizing search. Discrete Appl. Math. 39, 207229.
[6] Marshall, A. W., Olkin, I. and Arnold, B.C. (2011). Inequalities: Theory of Majorization and Its Applications. 2nd. edn. Springer-Verlag, New York.
[7] Neal, P. (2008). The generalised coupon collector problem. J. Appl. Prob. 45, 621629.
[8] Rubin, H. and Zidek, J. (1965). A waiting time distribution arising from the coupon collector's problem. Tech. Rep. 107, Department of Statistics, Stanford University.
[9] Sericola, B. (2013). Markov Chains: Theory, Algorithms and Applications. Wiley-ISTE, London.

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New Results on a Generalized Coupon Collector Problem Using Markov Chains

  • Emmanuelle Anceaume (a1), Yann Busnel (a2) and Bruno Sericola (a3)

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