Advances in Applied Probability

The coupon collector's problem revisited: asymptotics of the variance

Aristides V. Doumas and Vassilis G. Papanicolaou

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We develop techniques for computing the asymptotics of the first and second moments of the number TN of coupons that a collector has to buy in order to find all N existing different coupons as N → ∞. The probabilities (occurring frequencies) of the coupons can be quite arbitrary. From these asymptotics we obtain the leading behavior of the variance V[TN] of TN (see Theorems 3.1 and 4.4). Then, we combine our results with the general limit theorems of Neal in order to derive the limit distribution of TN (appropriately normalized), which, for a large class of probabilities, turns out to be the standard Gumbel distribution. We also give various illustrative examples.

Article information

Adv. in Appl. Probab., Volume 44, Number 1 (2012), 166-195.

First available in Project Euclid: 8 March 2012

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F99: None of the above, but in this section

Coupon collector's problem higher asymptotics limit distribution


Doumas, Aristides V.; Papanicolaou, Vassilis G. The coupon collector's problem revisited: asymptotics of the variance. Adv. in Appl. Probab. 44 (2012), no. 1, 166--195. doi:10.1239/aap/1331216649.

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  • Apostol, T. M. (1976). Introduction to Analytic Number Theory. Springer, New York.
  • Bender, C. M. and Orszag, S. A. (1999). Advanced Mathematical Methods for Scientists and Engineers. Springer, New York.
  • Boneh, A. and Hofri, M. (1997). The coupon-collector problem revisited–-a survey of engineering problems and computational methods. Commun. Statist. Stoch. Models 13, 39–66.
  • Boneh, S. and Papanicolaou, V. G. (1996). General asymptotic estimates for the coupon collector problem. J. Comput. Appl. Math. 67, 277–289.
  • Boros, G. and Moll, V. (2004). Irresistible Integrals. Cambridge University Press.
  • Brayton, R. K. (1963). On the asymptotic behavior of the number of trials necessary to complete a set with random selection. J. Math. Anal. Appl. 7, 31–61.
  • Durrett, R. (2005). Probability: Theory and Examples, 3rd edn. Cambridge University Press.
  • Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. I, John Wiley, New York.
  • Flajolet, P., Gardy, D. and Thimonier, L. (1992). Birthday paradox, coupon collectors, caching algorithms and self-organizing search. Discrete Appl. Math. 39, 207–229.
  • Hildebrand, M. V. (1993). The birthday problem. Amer. Math. Monthly 100, 643.
  • Holst, L., Kennedy, J. E. and Quine, M. P. (1988). Rates of Poisson convergence for some coverage and urn problems using coupling. J. Appl. Prob. 25, 717–724.
  • Neal, P. (2008). The generalised coupon collector problem. J. Appl. Prob. 45, 621–629.
  • Ross, S. (2006). A First Course in Probability, 7th edn. Pearson Prentice Hall.
  • Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill, New York.