The origin of power-law behavior (also known variously as Zipf’s law) has been a topic of debate in the scientific community for more than a century. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. In a highly cited article, Mark Newman [Contemp. Phys. 46 (2005) 323–351] reviewed some of the empirical evidence for the existence of power-law forms, however underscored that even though many distributions do not follow a power law, quite often many of the quantities that scientists measure are close to a Zipf law, and hence are of importance. In this paper we engage a variant of Zipf’s law with a general urn problem. A collector wishes to collect m complete sets of N distinct coupons. The draws from the population are considered to be independent and identically distributed with replacement, and the probability that a type-j coupon is drawn is denoted by p$$, j = 1, 2, …, N. Let T$$(N) the number of trials needed for this problem. We present the asymptotics for the expectation (five terms plus an error), the second rising moment (six terms plus an error), and the variance of T$$(N) (leading term) as N →∞, when
Moreover, we prove that T$$(N) (appropriately normalized) converges in distribution to a Gumbel random variable. These “log-Zipf” classes of coupon probabilities are not covered by the existing literature and the present paper comes to fill this gap. In the spirit of a recent paper of ours [ESAIM: PS 20 (2016) 367–399] we enlarge the classes for which the Dixie cup problem is solved w.r.t. its moments, variance, distribution.
Mots-clés : Generalized Zipf law, Urn problems, coupon collector’s problem, double Dixie cup problem, Gumbel distribution, Laplace method for integrals – determination of higher order terms, Eulerian logarithmic integral
@article{PS_2020__24_1_275_0,
author = {Doumas, Aristides V. and Papanicolaou, Vassilis G.},
title = {The logarithmic {Zipf} law in a general urn problem},
journal = {ESAIM: Probability and Statistics},
pages = {275--293},
publisher = {EDP-Sciences},
volume = {24},
year = {2020},
doi = {10.1051/ps/2020011},
mrnumber = {4090339},
zbl = {1434.60069},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2020011/}
}
TY - JOUR AU - Doumas, Aristides V. AU - Papanicolaou, Vassilis G. TI - The logarithmic Zipf law in a general urn problem JO - ESAIM: Probability and Statistics PY - 2020 SP - 275 EP - 293 VL - 24 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2020011/ DO - 10.1051/ps/2020011 LA - en ID - PS_2020__24_1_275_0 ER -
%0 Journal Article %A Doumas, Aristides V. %A Papanicolaou, Vassilis G. %T The logarithmic Zipf law in a general urn problem %J ESAIM: Probability and Statistics %D 2020 %P 275-293 %V 24 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2020011/ %R 10.1051/ps/2020011 %G en %F PS_2020__24_1_275_0
Doumas, Aristides V.; Papanicolaou, Vassilis G. The logarithmic Zipf law in a general urn problem. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 275-293. doi : 10.1051/ps/2020011. http://www.numdam.org/articles/10.1051/ps/2020011/
[1] and , Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer-Verlag, New York (1999). | DOI | MR | Zbl
[2] and , General asymptotic estimates for the coupon collector problem. J. Comput. Appl. Math. 67 (1996) 277–289. | DOI | MR | Zbl
[3] , On the asymptotic behavior of the number of trials necessary to complete a set with random selection. Math. Anal. Appl. 7 (1963) 31–61. | DOI | MR | Zbl
[4] , Asymptotic Methods in Analysis, second edition. North Holland, Amsterdam (1961). | MR | Zbl
[5] and , A Bayesian peek into Feller volume I, in Vol. 64, No. 3, In Memory of D. Basu, Part 2 (2002) 820–841. | MR | Zbl
[6] and , The Coupon Collector’s problem revisited: asymptotics of the variance. Adv. Appl. Prob. 44 (2012) 166–195. | DOI | MR | Zbl
[7] and , Asymptotics of the rising moments for the Coupon Collector’s Problem. Electron. J. Probab. 18 (2013) 41. | DOI | MR | Zbl
[8] and , The Coupon Collector’s problem revisited: generalizing the double dixie cup problem of Newman and Shepp. ESAIM: PS 20 (2016) 367–399. | DOI | Numdam | MR | Zbl
[9] and , Uniform versus Zipf distribution in a mixing collection process. Stat. Probab. Lett. 155 (2019) 108559. | DOI | MR | Zbl
[10] and , Sampling from a mixture of different groups of coupons. Preprint (2018). | arXiv | MR
[11] , Probability: Theory and Examples, Third Edition, Duxbury Advanced Series, Brooks/Cole—Thomson Learning. Belmont, CA, USA (2005). | Zbl
[12] and , On a classical problem of probability theory. Magyar. Tud. Akad. Mat. Kutató Int. Közl. 6 (1961) 215–220. | MR | Zbl
[13] , An Introduction to Probability Theory and Its Applications, Vol. I & II. John Wiley & Sons, Inc., New York (1966). | MR | Zbl
[14] and , Analytic Combinatorics. Cambridge University Press, 1st Ed. (2009). | DOI | MR | Zbl
[15] , and , Birthday paradox, coupon collectors, caching algorithms and self-organizing search, Discr. Appl. Math. 39 (1992) 207–229. | DOI | MR | Zbl
[16] , , Collectors’, Occupancy and other classical Urn problems. Int. Stat. Rev. 54 (1986) 15–27. | DOI | MR | Zbl
[17] and , Scaling laws predict global microbial diversity. Proc. Natl. Acad. Sci. USA 113 (2016) 5970–5975. | DOI
[18] , Pólya urn models. CRC Press, New York (2008). | DOI | MR | Zbl
[19] , and , Inequalities: Theory of Majorization and Its Applications. Springer, 2nd ed. (2009). | MR | Zbl
[20] , The Generalised Coupon Collector Problem. J. Appl. Prob. 45 (2008) 621–629. | DOI | MR | Zbl
[21] , Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46 (2005) 323–351. | DOI
[22] and , The double Dixie cup problem. Am. Math. Monthly 67 (1960) 58–61. | DOI | MR | Zbl
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