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

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.