Journal of Theoretical Probability

, Volume 28, Issue 4, pp 1311–1336 | Cite as

Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems

Article

Abstract

Using the one-to-one correspondence between copulas and special Markov kernels three strong metrics on the class \(\mathcal {C}_\rho \) of \(\rho \)-dimensional copulas with \(\rho \ge 3\) are studied. Being natural extensions of the two-dimensional versions introduced by Trutschnig (J Math Anal Appl 384:690–705, 2011), these metrics exhibit various good properties. In particular, it can be shown that the resulting metric spaces are separable and complete, which, as by-product, offers a simple separable and complete metrization of the so-called \(\partial \)-convergence studied by Mikusinski and Taylor (Ann Polon Math 96:75–95, 2009, Metrika 72:385–414, 2010). As an additional consequence of completeness, it is proved that the construction of singular copulas with fractal support via special Iterated Function Systems also converges with respect to any of the three introduced metrics. Moreover, the interrelation with the uniform metric \(d_\infty \) is studied and convergence with respect to \(d_\infty \) is characterized in terms of level-set and endograph convergence with respect to the Hausdorff metric.

Keywords

Copula Stochastic measure Markov kernel Iterated Function System Level set Endograph 

Mathematics Subject Classification (2010)

62H20 60E05 28A80 

Notes

Acknowledgments

The first author gratefully acknowledges the support of Grant MTM2011-22394 from the Spanish Ministry of Science and Innovation.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Grupo de Investigación de Análisis MatemáticoUniversidad de AlmeríaAlmeríaSpain
  2. 2.Department for MathematicsUniversity SalzburgSalzburgAustria

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