Journal of Applied Probability

Generalized coupon collection: the superlinear case

R. T. Smythe

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider a generalized form of the coupon collection problem in which a random number, S, of balls is drawn at each stage from an urn initially containing n white balls (coupons). Each white ball drawn is colored red and returned to the urn; red balls drawn are simply returned to the urn. The question considered is then: how many white balls (uncollected coupons) remain in the urn after the kn draws? Our analysis is asymptotic as n → ∞. We concentrate on the case when kn draws are made, where kn / n → ∞ (the superlinear case), although we sketch known results for other ranges of kn. A Gaussian limit is obtained via a martingale representation for the lower superlinear range, and a Poisson limit is derived for the upper boundary of this range via the Chen-Stein approximation.

Article information

J. Appl. Probab., Volume 48, Number 1 (2011), 189-199.

First available in Project Euclid: 15 March 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G42: Martingales with discrete parameter
Secondary: 05A05: Permutations, words, matrices 60C05: Combinatorial probability

Urn model martingale occupancy problem coupon collection central limit theorem Poisson limit


Smythe, R. T. Generalized coupon collection: the superlinear case. J. Appl. Probab. 48 (2011), no. 1, 189--199. doi:10.1239/jap/1300198144.

Export citation


  • Adler, I. and Ross, S. M. (2001). The coupon subset collection problem. J. Appl. Prob. 38, 737–746.
  • Arratia, R., Goldstein, L. and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5, 403–434.
  • Békéssy, A. (1963). On classical occupancy problems. I. Magyar Tud. Akad. Mat. Kuktató Int. Közl. 8, 59–71.
  • Chistyakov, V. P. (1964). On the calculation of the power of the test of empty boxes. Theory Prob. Appl. 9, 648–653.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications. Academic Press, New York.
  • Holst, L. (1971). Limit theorems for some occupancy and sequential occupancy problems. Ann. Math. Statist. 42, 1671–1680.
  • Ivchenko, G. I. (1998). How many samples does it take to see all of the balls in an urn? Math. Notes 64, 49–54.
  • Kobza, J. E., Jacobson, S. H. and Vaughan, D. E. (2007). A survey of the coupon collector's problem with random sample sizes. Methodology Comput. Appl. Prob. 9, 573–584.
  • Kolchin, V. F., Sevast'yanov, B. A. and Chistyakov, V. P. (1978). Random Allocations. Winston, Washington, DC.
  • Mahmoud, H. M. (2010). Gaussian phases in generalized coupon collection. Adv. Appl. Prob. 42, 994–1012.
  • Mikhaĭlov, V. (1977). A Poisson limit theorem in the scheme of group disposal of particles. Theory Prob. Appl. 22, 152–156.
  • Pólya, G. (1930). Eine Wahrscheinlichkeitsaufgabe zur Kunderwerbung. Z. Angew. Math. Mech. 10, 96–97.
  • Rényi, A. (1962). Three new proofs and a generalization of a theorem of Irving Weiss. Magyar Tud. Akad. Mat. Kutató Int. Közl. 7, 203–214.
  • Rosén, B. (1969). Asymptotic normality in a coupon collector's problem. Z. Wahrscheinlichkeitsth. 13, 256–279.
  • Sellke, T. (1995). How many i.i.d. samples does it take to see all the balls in a box? Ann. Appl. Prob. 5, 294–309.
  • Stadje, W. (1990). The collector's problem with group drawings. Adv. Appl. Prob. 22, 866–882.
  • Weiss, I. (1958). Limiting distributions in some occupancy problems. Ann. Math. Statist. 29, 878–884.
  • Von Mises, R. (1939). Über aufteilungs- und besetzungs-Wahrscheinlichkieten. Revu de la Faculté des Sciences de l'Université d'Istanbul, Vol. 4, pp. 145–163.
  • Zubkov, A. M. and Mikhaĭlov, V. G. (1974). Limit distributions of random variables associated with long duplications in a sequence of independent trials. Theory Prob. Appl. 19, 172–179.