Sampling from a Mixture of Different Groups of Coupons

作     者：Aristides V.DOUMAS Vassilis G.PAPANICOLAOU Aristides V. DOUMAS;Vassilis G. PAPANICOLAOU

作者机构：Department of MathematicsNational Technical University of AthensZografou Campus15780AthensGreece 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2020年第36卷第12期

页      面：1357-1383页

核心收录：

学科分类：02[经济学] 0202[经济学-应用经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070103[理学-概率论与数理统计] 0701[理学-数学] 

主　　题：Coupon collector problems urn problems asymptotics limiting distribution Gumbel distribution 

摘      要：A collector samples coupons with replacement from a pool containing g uniform groups of coupons,whereuniform groupmeans 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 *** 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{T1T2}as the number of coupons grows to infinity in a certain ***,we focus on T:=T1∨T2,*** 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.