On differentiability and mass distributions of topologically typical multivariate Archimedean copulas

TL;DR

This paper investigates the irregularity and mass distribution properties of multivariate Archimedean copulas, revealing that high-order derivatives can be pathological and that typical copulas tend to have degenerate discrete components rather than absolute continuity.

Contribution

It demonstrates the existence of irregular high-order derivatives in Archimedean copulas and characterizes their mass distributions, showing typical copulas are not absolutely continuous.

Findings

High-order derivatives of Archimedean copulas can be pathological.

The mass distribution of copulas relates closely to Williamson measures.

Typical copulas tend to have degenerated discrete components rather than absolute continuity.

Abstract

Copulas, in particular Archimedean copulas are commonly viewed as analytically nice and regular objects. Motivated by a recently established result sta\-ting that the first partial derivatives of bivariate copulas can exhibit surprisingly pathological behavior, we focus on the class of $d$-dimensional Archimedean copulas denoted by $\mathcal{C}_{ar}^d$ and show that partial derivatives of order $(d-1)$ can be sur\-pri\-singly irregular as well. In fact, we prove the existence of Archimedean copulas $C \in \mathcal{C}_{ar}^d$ whose $(d-1)$-st order partial derivatives are pathological in the sense that for almost every $\mathbf{x} \in [0,1]^{d-1}$ the derivative $\partial_1...\partial_{d-1}C(\mathbf{x},y)$ does not exist on a dense set of $y \in (0,1)$. \\ Since the existence of mixed partial derivatives of order $(d-1)$ of a copula $C$ is closely related to the existence of a discrete component, we also study mass distributions of Archimedean copulas. Building upon the interplay between Archimedean copulas and so-called Williamson measures we show that absolute continuity, discreteness and singularity of the Williamson measure propagates to the associated Archimedean copula and vice versa. Moreover, we prove the fact that the sub-family of $\mathcal{C}_{ar}^d$ consisting of copulas whose absolutely continuous, discrete and singular component have full support is dense in $\mathcal{C}_{ar}^d$. \\ Finally, viewing $\mathcal{C}_{ar}^d$ in the light of Baire categories, we show that, in contrast to the space of bivariate copulas, a topologically typical $d$-dimensional Archimedean copula $C$ is not absolutely continuous but has degenerated discrete component, implying that pathological elements are rare in $\mathcal{C}_{ar}^d$.