Convex lineability in copula and quasi-copula sets

TL;DR

This paper explores the large algebraic structures within various classes of copulas and quasi-copulas, focusing on convex lineability and spaceability, revealing which families contain extensive linearly independent sets.

Contribution

It introduces the concept of convex-spaceability in copula sets and characterizes when these classes contain large algebraic structures, advancing the understanding of their geometric properties.

Findings

Certain copula families contain continuum-sized linearly independent sets.

Convex lineability is established for some classes, while convex spaceability remains open for others.

The paper distinguishes between classes with and without maximal convex lineability.

Abstract

In this paper, we investigate several subsets of $n$-copulas and $n$-quasi-copulas from the perspective of convex-lineability and the recently introduced concept of convex-spaceability. Our purpose is to determine when such families contain extremely large algebraic structures, namely linearly independent sets of cardinality of the continuum whose convex hull, and in some cases a closed convex linearly independent subset, remain entirely inside the class under study. These include the families of asymmetric copulas, copulas with maximal asymmetric measure, and proper $n$-quasi-copulas, among others. In contrast, for several other natural classes of copulas we show that (maximal) convex lineability holds while convex spaceability remains an open problem.