zbMATH — the first resource for mathematics

Convergence of point processes associated with coupon collector’s and Dixie cup problems. (English) Zbl 1422.60082
Summary: We prove that, in the coupon collector’s problem, the point processes given by the times of $$r^{\mathrm{th}}$$ arrivals for coupons of each type, centered and normalized in a proper way, converge toward a non-homogeneous Poisson point process. This result is then used to derive some generalizations and infinite-dimensional extensions of classical limit theorems on the topic.

MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F17 Functional limit theorems; invariance principles
Full Text:
References:
 [1] E. Anceaume, Y. Busnel, and B. Sericola. New results on a generalized coupon collector problem using Markov chains. J. Appl. Probab., 52(2):405-418, 2015. · Zbl 1327.60135 [2] A. D. Barbour, L. Holst, and S. Janson. Poisson approximation. Oxford: Clarendon Press, 1992. · Zbl 0746.60002 [3] L. E. Baum and P. Billingsley. Asymptotic distributions for the coupon collector’s problem. Ann. Math. Statist., 36(6):1835-1839, 1965. · Zbl 0227.62010 [4] Yu. V. Bolotnikov. The convergence of the variables $$\mu _r(n)$$ to Gaussian and Poisson processes in the classical ball problem. Theory Probab. Appl., 13:39-51, 1968. · Zbl 0267.60008 [5] A. V. Doumas and V. G. Papanicolaou. Asymptotics of the rising moments for the coupon collector’s problem. Electron. J. Probab., 18:15, 2013. · Zbl 1283.60035 [6] A. V. Doumas and V. G. Papanicolaou. Some new aspects for the random coupon collector’s problem. ALEA, Lat. Am. J. Probab. Math. Stat., 11(1):197-208, 2014. · Zbl 1290.60010 [7] P. Erdős and A. Rényi. On a classical problem of probability theory. Publ. Math. Inst. Hung. Acad. Sci., Ser. A, 6:215-220, 1961. [8] M. Ferrante and M. Saltalamacchia. The coupon collector’s problem. Materials matemàtics, pages 01-35, 2014. [9] J. C. Fu and W.-C. Lee. On coupon collector’s and Dixie cup problems under fixed and random sample size sampling schemes. Ann. Inst. Stat. Math., 69(5):1129-1139, 2017. · Zbl 1383.60012 [10] L. Glavaš and P. Mladenović. New limit results related to the coupon collector’s problem. Stud. Sci. Math. Hung., 55(1):115-140, 2018. [11] L. Holst. On sequential occupancy problems. J. Appl. Probab., 18:435-442, 1981. · Zbl 0461.60028 [12] L. Holst. On birthday, collectors’, occupancy and other classical urn problems. Int. Stat. Rev., 54:15-27, 1986. · Zbl 0594.60014 [13] L. Holst. Extreme value distributions for random coupon collector and birthday problems. Extremes, 4(2):129-145, 2001. · Zbl 1003.60052 [14] O. Kallenberg. Random measures, theory and applications. Cham: Springer, 2017. · Zbl 1376.60003 [15] V. F. Kolchin, B. A. Sevast’yanov, and V. P. Chistyakov. Random allocations. New York etc.: John Wiley & Sons, 1978. [16] G. Last and M. Penrose. Lectures on the Poisson process. Cambridge: Cambridge University Press, 2017. · Zbl 1392.60004 [17] M. Mayer and I. Molchanov. Limit theorems for the diameter of a random sample in the unit ball. Extremes, 10(3):129-150, 2007. · Zbl 1164.62003 [18] P. Mladenović. Limit distributions for the problem of collecting pairs. Bernoulli, 14(2):419-439, 2008. [19] P. Neal. The generalised coupon collector problem. J. Appl. Probab., 45(3):621-629, 2008. · Zbl 1151.60315 [20] M. Penrose. Random geometric graphs. Oxford: Oxford University Press, 2003. · Zbl 1029.60007 [21] S. I. Resnick. Extreme values, regular variation, and point processes. New York etc.: Springer-Verlag, 1987. · Zbl 0633.60001 [22] S. I. Resnick. Heavy-tail phenomena. Probabilistic and statistical modeling. New York, NY: Springer, 2007. · Zbl 1152.62029 [23] N. Ross. Fundamentals of Stein’s method. Probab. Surv., 8:210-293, 2011. · Zbl 1245.60033 [24] E. Samuel-Cahn. Asymptotic distributions for occupancy and waiting time problems with positive probability of falling through the cells. Ann. Probab., 2(3):515-521, 1974. · Zbl 0285.60005
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.