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Convex lineability in copula and quasi-copula sets

Enrique de Amo, Juan Fernández-Sánchez, David García-Fernández, Manuel Úbeda-Flores

TL;DR

This work investigates how large algebraic and geometric structures can be embedded inside nonlinear families of copulas and quasi-copulas using convex-lineability and convex-spaceability. By constructing explicit, continuum-sized families and preserving convex hulls, the authors demonstrate maximal convex lineability for several classes (e.g., fractal-support copulas, Lipschitz-bounded families) and maximal spaceability for others (notably asymmetric and maximal-asymmetry copulas, and proper quasi-copulas with fixed diagonals). They also show that certain natural convergence-based sets of sequences of copulas exhibit maximal convex-lineable structure, while spaceability remains open in some cases, highlighting subtle interactions between ordinal-sum constructions and topological closure. Overall, the results reveal rich internal structure in dependence-modeling objects, with practical implications for constructing robust, convexly closed families of copulas and quasi-copulas for applications in finance, hydrology, and risk modeling.

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.

Convex lineability in copula and quasi-copula sets

TL;DR

This work investigates how large algebraic and geometric structures can be embedded inside nonlinear families of copulas and quasi-copulas using convex-lineability and convex-spaceability. By constructing explicit, continuum-sized families and preserving convex hulls, the authors demonstrate maximal convex lineability for several classes (e.g., fractal-support copulas, Lipschitz-bounded families) and maximal spaceability for others (notably asymmetric and maximal-asymmetry copulas, and proper quasi-copulas with fixed diagonals). They also show that certain natural convergence-based sets of sequences of copulas exhibit maximal convex-lineable structure, while spaceability remains open in some cases, highlighting subtle interactions between ordinal-sum constructions and topological closure. Overall, the results reveal rich internal structure in dependence-modeling objects, with practical implications for constructing robust, convexly closed families of copulas and quasi-copulas for applications in finance, hydrology, and risk modeling.

Abstract

In this paper, we investigate several subsets of -copulas and -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 -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.
Paper Structure (15 sections, 10 theorems, 68 equations, 4 figures)

This paper contains 15 sections, 10 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Copula theory
  4. Convex-lineability
  5. Convex-lineability in copula and quasi-copula sets
  6. Copulas with fractal support
  7. Asymmetric copulas
  8. Copulas with maximum asymmetric measure
  9. $1$-$p$-Lipschitz copulas that are not $1$-$p'$-Lipschitz for any $p'>p$
  10. Proper $n$-quasi-copulas
  11. Proper $2$-quasi-copulas with given diagonal section
  12. Sequences of copulas
  13. Sequences of copulas converging in $d_\infty$ but not converging in $D_1$
  14. Sequences of copulas converging in $D_1$ but not converging in total variation
  15. Conclusion

Key Result

Theorem 1

Let ${\mathbf X}=(X_1,X_2,\ldots,X_n)$ be a random $n$-vector with joint distribution function $H$ and one-dimensional marginal distributions $F_1,F_2,\ldots,F_n$. Then there exists an $n$-copula, $C$, uniquely determined on $\times_{i=1}^n Range(F_i)$, such that for all ${\mathbf x}\in[-\infty,+\infty]^n$. If all the marginals $F_i$ are continuous then the $n$-copula $C$ is unique.

Figures (4)

  • Figure 1: Situation described in the proof of Theorem \ref{['th:s2']}.
  • Figure 2: Support of the copula $C_{(c)}$.
  • Figure 3: Support of the copula $D_{(a)}$.
  • Figure 4: Support of the proper quasi-copula $Q_C$ given by \ref{['eq:QC']} for $C=M^2$, with red indicating negative probability mass and blue indicating positive probability mass.

Theorems & Definitions (30)

  • Definition 1
  • Theorem 1: Sklar
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 20 more