Characterizing Schur-concave commutative copulas as the closure of associative ones

Abstract

Let ${\mathcal{C}}_a$ denote the class of associative copulas, and let ${\overline{\mathcal{C}}}_a$ be the closure, in the uniform metric $d_{\infty }$, of the convex hull of ${\mathcal{C}}_a$ . It is known that ${\mathcal{C}}_a\subseteq {\mathcal{C}}_{SC}$, the class of Schur-concave commutative copulas. We prove the reverse inclusion, establishing ${\overline{\mathcal{C}}}_a={\mathcal{C}}_{SC}$.

arXiv commentThe author wish to withdraw this manuscript due to an error discovered which could invalidate the main conclusion of the work. The paper requires a complete revision