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 T_j be the number of trials needed to detect Group j, namely to collect all M_j coupons belonging to it at least once. We first derive formulas for the probabilities P {T₁ < … < T_g} and P {T₁=Λ_j=1^g T_j}. 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.