Generalized coupon collection: the superlinear case

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 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.