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

TL;DR

This paper proves that the closure of the convex hull of associative copulas equals the class of Schur-concave commutative copulas, establishing a key characterization in copula theory.

Contribution

It demonstrates that the Schur-concave commutative copulas are exactly the closure of the convex hull of associative copulas in the uniform metric.

Findings

Established the equality _{SC} = \u001C_a closure in the uniform metric.

Provided a characterization linking associative and Schur-concave copulas.

Enhanced understanding of the structure of copula classes.

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 $\overline{\mathcal{C}}_a = \mathcal{C}_{SC}$.