Doumas, Aristides V.; Papanicolaou, Vassilis G. Sampling from a mixture of different groups of coupons. (English) Zbl 1462.60011 Acta Math. Sin., Engl. Ser. 36, No. 12, 1357-1383 (2020). Summary: 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 \(j=1,\dots,g\), 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 \(P\{T_1< \dots< T_g\}\) and \(P\left\{{{T_1}=\wedge_{j=1}^g\,{T_j}}\right\}\). After that, without severe loss of generality, we restrict ourselves to the case \(g=2\) and compute the asymptotics of \(P\{T_1 < T_2\}\) as the number of coupons grows to infinity in a certain manner. Then, we focus on \(T:=T_1 \vee T_2\), 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. Cited in 1 Document MSC: 60C05 Combinatorial probability 60F05 Central limit and other weak theorems 60F99 Limit theorems in probability theory 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) Keywords:coupon collector problems; urn problems; asymptotics; limiting distribution; Gumbel distribution PDFBibTeX XMLCite \textit{A. V. Doumas} and \textit{V. G. Papanicolaou}, Acta Math. Sin., Engl. Ser. 36, No. 12, 1357--1383 (2020; Zbl 1462.60011) Full Text: DOI arXiv References: [1] Bender, C. M.; Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory (1999), New York: Springer-Verlag, New York · Zbl 0938.34001 [2] Boros, G.; Moll, V. H., Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals (2004), Edinburgh: Cambridge University Press, Edinburgh · Zbl 1090.11075 [3] Doumas, A. V.; Papanicolaou, V. G T. C C. P r., Asymptotics of the variance, Adv. Appl. Prob., 44, 1, 166-195 (2012) · Zbl 1260.60035 [4] Doumas, A. V.; Papanicolaou, V. G., Asymptotics of the rising moments for the Coupon Collector’s Problem, Electron. J. Probab., 18, 1-15 (2012) · Zbl 1283.60035 [5] Doumas, A. V.; Papanicolaou, V. G., Uniform versus Zipf distribution in a mixing collection process, Stat. and Probab. Let., 155, Article no. 108559 (2019) · Zbl 1423.60042 [6] Doumas, A. V.; Papanicolaou, V. G., The Coupon Collector’s Problem Revisited: Generalizing the Double Dixie Cup Problem of Newman, ESAIM Probab. Stat., 20, 367-399 (2016) · Zbl 1355.60029 [7] Erdős, P.; Rényi, A., On a classical problem of probability theory, Magyar. Tud. Akad. Mat. Kutató Int. Közl., 6, 215-220 (1961) · Zbl 0102.35201 [8] Ross, S. M.: Private communication, 2017 [9] Ross, S. M., Introduction to Probability Models (2010), Burlington, MA: Elsevier Inc., Burlington, MA · Zbl 1184.60002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.