Sampling from a Mixture of Different Groups of Coupons

A collector samples coupons with replacement from a pool containing g uniform groups of coupons,where"uniform group"means that all coupons in the group are equally likely to occur(while coupons of different groups have different probabilities to occur).For each j=1,...,g,let Tj be the number of trials needed to detect Group j,namely to collect all Mj coupons belonging to it at least once.We first derive formulas for the probabilities P{T1<···<Tg}and P{T1=∧^gj=1Tj}.After that,without severe loss of generality,we restrict ourselves to the case g=2 and compute the asymptotics of P{T1<T2}as the number of coupons grows to infinity in a certain manner.Then,we focus on T:=T1∨T2,i.e.the number of trials needed to collect all coupons of the pool(at least once),and determine the asymptotics of E[T]and V[T],as well as the limiting distribution of T(appropriately normalized)as the number of coupons becomes large.