On Topologically Typical Bivariate Extreme Value Copulas

Motivated by the fact that (in the sense of Baire categories and working with the uniform distance \(d_\infty \) ) typical bivariate copulas are completelyDietrich, N.P.Trutschnig, W. dependent, we prove that even the subclass of all mutually completely dependent copulas with full support are typical (co-meager). Additionally, considering the subclass of Extreme Value copulas, working with so called Pickands dependence measures, i.e., univariate probability measures with expected value \(\frac{1}{2}\) , allows not only to determine the support of the Extreme Value copula via the afore-mentioned measure, but also, working with Markov-kernels, to characterize the discrete component of the Extreme Value copula by the point masses of the Pickands dependence measure. Establishing a homeomorphism between the spaces of all Pickands dependence measures and the family of Extreme Value copulas and using Markov-kernels, we derive the surprising result that, in contrast to the family of all bivariate copulas, a typical Extreme Value copula has degenerated discrete component, is not absolutely continuous and has full support.