On a Generalisation of the Coupon Collector Problem

Article Type

Research Article

Publication Title

Journal of Theoretical Probability

Abstract

We consider a generalisation of the classical coupon collector problem. We define a super-coupon to be any s-subset of a universe of n coupons. In each round, a random r-subset from the universe is drawn and all its s-subsets are marked as collected. We show that the time to collect all super-coupons is rs-1nslogns[(1+o(1))] on average and has a Gumbel limit after a suitable normalisation. In a similar vein, we show that for any α∈(0,1), the expected time to collect (1-α)-proportion of all super-coupons is rs-1nslog(1α)[(1+o(1))]. The r=s case of this model is equivalent to the classical coupon collector model. We also consider a temporally dependent model where the r-subsets are drawn according to the following Markovian dynamics: the r-subset at round k+1 is formed by replacing a random coupon from the r-subset drawn at round k with another random coupon from outside this r-subset. We link the time it takes to collect all super-coupons in the r=s case of this model to the cover time of random walk on a certain finite regular graph and conjecture that in general, it takes rsrs-1nslogns[(1+o(1))] time on average to collect all super-coupons.

DOI

10.1007/s10959-025-01417-w

Publication Date

9-1-2025

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