Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-17T19:51:53.617Z Has data issue: false hasContentIssue false
  • English
  • Français

Gaussian phases in generalized coupon collection

Part of: Limit theorems Stochastic processes Combinatorial probability Combinatorics

Published online by Cambridge University Press:  01 July 2016

Hosam M. Mahmoud
Hosam M. Mahmoud*
Affiliation:
The George Washington University
*
∗ Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider a generalized coupon collection problem in which a customer repeatedly buys a random number of distinct coupons in order to gather a large number n of available coupons. We address the following question: How many different coupons are collected after k = kn draws, as n → ∞? We identify three phases of kn: the sublinear, the linear, and the superlinear. In the growing sublinear phase we see o(n) different coupons, and, with true randomness in the number of purchases, under the appropriate centering and scaling, a Gaussian distribution is obtained across the entire phase. However, if the number of purchases is fixed, a degeneracy arises and normality holds only at the higher end of this phase. If the number of purchases have a fixed range, the small number of different coupons collected in the sublinear phase is upgraded to a number in need of centering and scaling to become normally distributed in the linear phase with a different normal distribution of the type that appears in the usual central limit theorems. The Gaussian results are obtained via martingale theory. We say a few words in passing about the high probability of collecting nearly all the coupons in the superlinear phase. It is our aim to present the results in a way that explores the critical transition at the ‘seam line’ between different Gaussian phases, and between these phases and other nonnormal phases.

Keywords

Urn modelrandom structuremartingalecentral limit theoremcoupon collection

MSC classification

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 05A05: Permutations, words, matrices
Secondary: 60G42: Martingales with discrete parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 994 - 1012
Copyright
Copyright © Applied Probability Trust 2010 

References

Adler, I. and Ross, S. M. (2001). The coupon subset collection problem. J. Appl. Prob. 38, 737–746.Google Scholar
Bender, E. A., Canfield, E. R. and McKay, B. D. (1997). The asymptotic number of labeled graphs with n vertices, q edges, and no isolated vertices. J. Combinatorial Theory A 80, 124–150.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications. Academic Press, New York.Google Scholar
Ivchenko, G. I. (1998). How many samples does it take to see all of the balls in an urn? Math. Notes 64, 49–54.Google Scholar
Johnson, B. C. and Sellke, T. M. (2010). On the number of i.i.d. samples required to observe all of the balls in an urn. Methodology Comput. Appl. Prob. 12, 139–154.CrossRefGoogle Scholar
Karr, A. F. (1993). Probability. Springer, New York.Google Scholar
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.Google Scholar
Kolchin, V. F., Sevastyanov, B. A. and Chistyakov, V. P. (1978). Random Allocations. John Wiley, New York.Google Scholar
Mikhaǐlov, V. G. (1977). A Poisson limit theorem in the scheme of group disposal of particles. Theory Prob. Appl. 22, 152–156.CrossRefGoogle Scholar
Mikhaǐlov, V. G. (1980). Asymptotic normality of the number of empty cells for group allocation of particles. Theory Prob. Appl. 25, 82–90.CrossRefGoogle Scholar
Pólya, G. (1930). Eine Wahrscheinlichkeitsaufgabe zur Kundenwerbung. Z. Angew. Math. Mech. 10, 96–97.Google Scholar
Sellke, T. M. (1995). How many i.i.d. samples does it take to see all the balls in a box? Ann. Appl. Prob. 5, 294–309.Google Scholar
Smythe, R. (2009). Phases in generalized coupon collection. Personal communication.Google Scholar
Stadje, W. (1990). The collector's problem with group drawings. Adv. Appl. Prob. 22, 866–882.Google Scholar
Vatutin, V. A. and Mikhaǐlov, V. G. (1982). Limit theorems for the number of empty cells in an equiprobable scheme for group allocation of particles. Theory Prob. Appl. 27, 734–743.CrossRefGoogle Scholar