Abstract
In [7] Fredricks et al. showed that certain Iterated Function Systems (IFS) can be used to construct copulas with fractal support. Since, firstly, the same construction also works with respect to the D 1-metric on the space of copulas which is much stronger than the uniform metric and, secondly, the star product of copulas is (jointly) continuous with respect to this metric the IFS approach can also be used to construct idempotent copulas with fractal support. The main result of the paper is that for each open interval I ⊆ [1,2] there exists an idempotent copula A such that the Hausdorff dimension of the support of A is contained in the interval I.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barnsley, M.F.: Fractals everywhere. Academic Press, Cambridge (1993)
Darsow, W.F., Nguyen, B., Olsen, E.T.: Copulas and Markov processes. Ill. J. Math. 36(4), 600–642 (1992)
Darsow, W.F., Olsen, E.T.: Characterization of idempotent 2-copulas. Note Mat. 30(1), 147–177 (2010)
Durante, F., Klement, E.P., Quesada-Molina, J., Sarkoci, J.: Remarks on Two Product-like Constructions for Copulas. Kybernetika 43(2), 235–244 (2007)
Durante, F., Sempi, C.: Copula theory: an introduction. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (eds.) Copula Theory and its Applications. Lecture Notes in Statistics–Proceedings, vol. 198, pp. 1–31. Springer, Berlin (2010)
Farahat, H.K.: The semigroup of doubly-stochastic matrices. Proc. Glasgow Mat., Ass. 7, 178–183 (1966)
Fredricks, G.A., Nelsen, R.B., Rodríguez-Lallena, J.A.: Copulas with fractal supports. Insur. Math. Econ. 37, 42–48 (2005)
Kallenberg, O.: Foundations of modern probability. Springer, Heidelberg (1997)
Klenke, A.: Probability Theory - A Comprehensive Course. Springer, Heidelberg (2007)
Lancaster, H.O.: Correlation and complete dependence of random variables. Ann. Math. Stat. 34, 1315–1321 (1963)
Nelsen, R.B.: An Introduction to Copulas. Springer, New York (2006)
Olsen, E.T., Darsow, W.F., Nguyen, B.: Copulas and Markov operators. In: Proceedings of the Conference on Distributions with Fixed Marginals and Related Topics. IMS Lecture Notes. Monograph Series, vol. 28, pp. 244–259 (1996)
Schwarz, S.: A Note on the Structure of the Semigroup of Doubly-Stochastic Matrices. Mathematica Slovaca 17(4), 308–316 (1967)
Sempi, C.: Conditional Expectations and Idempotent Copulae. In: Cuadras, C.M., et al. (eds.) Distributions with Given Marginals and Statistical Modelling, pp. 223–228. Kluwer, Netherlands (2002)
Trutschnig, W.: On a strong metric on the space of copulas and its induced dependence measure. J. Math. Anal. Appl. 384, 690–705 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Trutschnig, W. (2012). Idempotent Copulas with Fractal Support. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances in Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 298. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31715-6_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-31715-6_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31714-9
Online ISBN: 978-3-642-31715-6
eBook Packages: Computer ScienceComputer Science (R0)