The clumsy coupon collector's problem

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 $p=ω(1/m)$, 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.