The closed convex hull of all associative copulas is a Z-set in the family of all bivariate Schur concave copulas, and related results

Abstract

Very recently, Problem 2.5 stated by Mesiar and Sarkoci in \cite{MeSa} and originally formulated by Durante was answered in the negative by Šeliga \cite{Sel}. The author showed that the closed convex hull of all bivariate associative copulas is a proper subset of the family of all Schur concave copulas, and even computed the exact uniform distance of his counterexamples to the closed convex hull (so that the Hausdorff distance between the two families is at least $\frac32-\sqrt{2}$). Using the fact that every bivariate associative copula can be represented as a (finite or countably infinite) ordinal sum of Archimedean copulas and the minimum copula $M$, and working with the Markov kernels of Archimedean copulas, we first give an alternative, simple proof of the strict inclusion. We do so by showing that convex combinations of associative copulas can assign only very little mass to \new{staircase-shaped unions of small squares covering} certain line segments parallel to the main diagonal, a result we later extend to so-called chains and staircases. Building on the first observation, we then go one step further and show that, from a topological perspective, the closed convex hull of all associative copulas is a very small subset of the Schur concave family, namely a so-called $Z$-set, and that the family of associative copulas is, in turn, a $Z$-set in its closed convex hull.