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Generalized Coupon Collection: The Superlinear Case

Part of: Limit theorems Stochastic processes Combinatorial probability Combinatorics

Published online by Cambridge University Press:  14 July 2016

R. T. Smythe
R. T. Smythe*
Affiliation:
Oregon State University
*
∗Postal address: Department of Statistics, Oregon State University, Corvallis, OR 97331-4606, USA. Email address: [email protected]
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Abstract

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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.

Keywords

Urn modelmartingaleoccupancy problemcoupon collectioncentral limit theoremPoisson limit

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60G42: Martingales with discrete parameter 05A05: Permutations, words, matrices 60C05: Combinatorial probability
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 189 - 199
Copyright
Copyright © Applied Probability Trust 2011 

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