The clumsy coupon collector's problem

TL;DR

This paper studies a generalized coupon collector problem where coupons can be lost, analyzing the time to collect all coupons under different probabilistic regimes as the number of coupons grows large.

Contribution

It establishes limit theorems for the collection time, describing its asymptotic behavior and limiting distributions across three regimes of the loss probability.

Findings

Gumbel limit for low loss probability regime

Exponential limit for high loss probability regime

Full characterization of the critical regime with birth-death process

Abstract

We consider a generalisation of the classical coupon collector's problem, in which at each time step a collector either receives a new copy of a randomly chosen coupon, or loses all their previously collected copies of that coupon. We consider the amount of time it takes this clumsy coupon collector to obtain the full set of $m$ coupons. We establish limit theorems as $m\to\infty$ for the clumsy coupon collection time, and describe the large $m$ asymptotics of its mean and variance. We identify three regimes, depending on how the probability of a clumsy update, $p$, scales with $m$. If $p=o(1/m)$, we obtain a Gumbel limit theorem, as is the case for the classical coupon collector. If $p=\omega(1/m)$, we instead show weak convergence to an exponential random variable. In the critical case, $p=c/m$, we give a full characterisation of the limiting distribution in terms of a birth-death process.