Generalized coupon collection: the superlinear case

Smythe, R. T.

J. Appl. Probab., Tome 48 (2011) no. 1, p. 189-199 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2011-03-15
Classification:  Urn model,  martingale,  occupancy problem,  coupon collection,  central limit theorem,  Poisson limit,  60F05,  60G42,  05A05,  60C05