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.

Approaching the coupon collector’s problem with group drawings via Stein’s method

“…for and , respectively. In [17] the author proves a central limit theorem for a generalised coupon collector’s problem allowing for random package sizes S in the lower superlinear regime, i.e. for and using a martingale representation.…”