The Generalised Coupon Collector Problem

Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y , from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distributionXwith the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the asymptotic distribution of X. The results are proved by studying the total number of coupons,W(t), not collected in t draws from the population and noting that P(Y ≤ t) = P(W(t) = 0). Two normalizations of Y are considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists for W(t). Finally, extensions to the K-coupon collector problem and the birthday problem are given.