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.


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


Zbl 1178.62056
Full Text: DOI


[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
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.