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Measure-preserving functions and the independence copula. (English) Zbl 1242.62042

Authors’ abstract: We solve a problem recently proposed by A. Kolesárová et al. [ibid. 5, No. 3, 325–339 (2008; Zbl 1178.62056)]. Specifically, we prove that a necessary and sufficient condition for a given copula to be an independence or product copula is for the pair of measure-preserving transformations representing the copula to be independent as random variables. We provide examples of such pairs for the well-known Cantor, Peano, and Hilbert curves. Moreover, a general constructive method is given for the representation of copulae in terms of measure-preserving transformations. In particular, we apply numbers representation systems to the study of self-similar copulae properties.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
28D05 Measure-preserving transformations
60E05 Probability distributions: general theory

Citations:

Zbl 1178.62056
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References:

[1] Billingsley, P.: Probability and Measure (3rd ed.), John Wiley & Sons, Inc., 1995 · Zbl 0822.60002
[2] Dajani, K.; and Kraaikamp, C.: Ergodic Theory of Numbers, The Mathematical Association of America Monographs 29, Washington, 2002 · Zbl 1033.11040
[3] Durante F., Klement E.P., Quesada-Molina J.J., Sarkoci P.: Remarks on two product-like constructions for copulas. Kybernetika (Prague) 43, 235–244 (2007) · Zbl 1136.60306
[4] Durante F., Sarkoci P., Sempi C.: Shuffles of copulas. J. Math. An. Appl. 352, 914–921 (2009) · Zbl 1160.60307 · doi:10.1016/j.jmaa.2008.11.064
[5] Edgar, G.A.: Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics, Springer. Berlin, Heidelberg, New York, 1990 · Zbl 0727.28003
[6] Edgar, G.A.: Integral, Probability and Fractal Measures, Springer Verlag, 1998 · Zbl 0893.28001
[7] Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications (2nd Ed.), Wiley, 1990 · Zbl 0689.28003
[8] Feller : Introduction to Probability Theory and Its Applications (3rd ed). John Wiley & Sons, New York (1968) · Zbl 0155.23101
[9] Fredricks G.A., Nelsen R.B, Rodríguez-Lallena J.A.: Copulas with fractal supports, Insurance Math. Econom. 37(1), 42–48 (2005) · Zbl 1098.60018
[10] Halmos P.R.: Ergodic Theory. Chelsea, New York (1956) · Zbl 0073.09302
[11] Hilbert D.: Ueber die stetige Abbildung einer Line auf ein Flächenstück. Mathematische Annalen 38, 459–460 (1891) · doi:10.1007/BF01199431
[12] Kolesárová A., Mesiar R, Sempi C.: Measure-preserving transformations, copula and compatibility. Mediterr. J. Math. 5, 325–339 (2008) · Zbl 1178.62056 · doi:10.1007/s00009-008-0153-2
[13] McClure, M.: The Hausdorff dimension of Hilbert’s coordinate functions, The Real Analysis Exchange, 24 (1998/99) 875-884 · Zbl 0967.28004
[14] Mikusiński P., Sherwood H., Taylor M.D.: Shuffles of Min. Stochastica 12, 61–72 (1992) · Zbl 0768.60017
[15] Nelsen R.B.: An introduction to copulas (2nd ed). Springer Series in Statistics, New York (2006) · Zbl 1152.62030
[16] Olsen, E.T.; Darsow, W.F.; and Nguyen, B.: Copulas and Markov operators, in: Distributions with Fixed Marginals and Related Topics, Seattle, WA, 1993, in: IMS Lecture Notes Monogr. Ser. 28 Inst. Math. Statist., Hayward, CA (1996) 244-259
[17] Peano G.: Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen 36(1), 157–160 (1890) · doi:10.1007/BF01199438
[18] Pollicott, M.; and Yuri, M.: Dynamical systems and ergodic theory, London Mathematical Society Student Texts, 40, Cambridge University Press, Cambridge, 1998 · Zbl 0897.28009
[19] Riesz F., Sz-Nagy B.: Leçons d’analyse fonctionnelle. Gauthier-Villars, Paris (1968)
[20] Rudin W.: Real and complex analysis (3rd ed). McGraw-Hill, New York (1966) · Zbl 0142.01701
[21] Sagan H.: Space-Filling Curves. Springer-Verlag, New York (1994) · Zbl 0806.01019
[22] Sklar A.: Fonctions de répartition á n dimensions et leurs marges. Publ. Inst. Statidt. Univ. Paris 8, 229–231 (1959) · Zbl 0100.14202
[23] Urbański M.: The Hausdorff dimension of the graphs of continuous self-affine functions. Proc. Amer. Math. Soc. 108(4), 921–930 (1990) · Zbl 0721.28004 · doi:10.2307/2047947
[24] Vitale, R.A.: Parametrizing doubly stochastic measures, in Distributions with Fixed Marginals and Related Topics, Seattle, WA, 1993, in: IMS Lecture Notes Monogr. Ser. 28, Inst. Math. Statist., Hayward, CA (1996) 358-364
[25] Walters, P.: An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1980 · Zbl 0475.28009
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