A scalar product for copulas

Abstract

We introduce a scalar product for n-dimensional copulas, based on the Sobolev scalar product for W 1,2 -functions. The corresponding norm has quite remarkable properties and provides a new geometric framework for copulas. We show that, in the bivariate case, it measures invertibility properties with respect to the ∗-product for copulas defined by Darsow et al. The unique copula of minimal norm is the null element for the ∗-multiplication, whereas the copulas of maximal norm are precisely the invertible elements.