Journal of Applied Probability

Generalized coupon collection: the superlinear case

R. T. Smythe

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Abstract

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

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

Dates
First available in Project Euclid: 15 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.jap/1300198144

Digital Object Identifier
doi:10.1239/jap/1300198144

Mathematical Reviews number (MathSciNet)
MR2809895

Zentralblatt MATH identifier
1213.60050

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

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

Citation

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


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