Generalized Coupon Collection: The Superlinear Case

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 que...

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Bibliographic Details
Published in:Journal of applied probability 2011-03, Vol.48 (1), p.189-199
Main Author: Smythe, R. T.
Format: Article
Language:English
Subjects:
05A05
60C05
60F05
60G42
Applications
Approximation
Asymptotic methods
Central limit theorem
coupon collection
Coupons
Determinism
Exact sciences and technology
Limit theorems
martingale
Martingales
Mathematical theorems
Mathematics
occupancy problem
Poisson limit
Probability and statistics
Probability distribution
Probability theory and stochastic processes
Random numbers
Random variables
Sample size
Sciences and techniques of general use
Statistics
Stochastic processes
Studies
Urn model
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Description
Summary: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 k n draws? Our analysis is asymptotic as n → ∞. We concentrate on the case when k n draws are made, where k n / n → ∞ (the superlinear case), although we sketch known results for other ranges of k n . 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.
ISSN:0021-9002
1475-6072
DOI:10.1239/jap/1300198144