On convergence of associative copulas and related results

Thimo M Kasper,Wolfgang Trutschnig,Sebastian Fuchs

doi:10.1515/demo-2021-0114

Abstract

AbstractTriggered by a recent article establishing the surprising result that within the class of bivariate Archimedean copulas 𝒞ardifferent notions of convergence - standard uniform convergence, convergence with respect to the metricD1, and so-called weak conditional convergence - coincide, in the current contribution we tackle the natural question, whether the obtained equivalence also holds in the larger class of associative copulas 𝒞a. Building upon the fact that each associative copula can be expressed as (finite or countably infinite) ordinal sum of Archimedean copulas and the minimum copulaMwe show that standard uniform convergence and convergence with respect toD1are indeed equivalent in 𝒞a. It remains an open question whether the equivalence also extends to weak conditional convergence. As by-products of some preliminary steps needed for the proof of the main result we answer two conjectures going back to Durante et al. and show that, in the language of Baire categories, when working withD1a typical associative copula is Archimedean and a typical Archimedean copula is strict.

Highlights

Various di erent notions of convergence in the family of bivariate copulas C have been considered in the literature: The standard uniform metric d∞ is probably the most common choice; since, d∞ is not capable of distinguishing independence and complete dependence the stronger metric D was introduced in [36]
Triggered by a recent article establishing the surprising result that within the class of bivariate Archimedean copulas Car di erent notions of convergence - standard uniform convergence, convergence with respect to the metric D, and so-called weak conditional convergence - coincide, in the current contribution we tackle the natural question, whether the obtained equivalence holds in the larger class of associative copulas Ca
Building upon the fact that each associative copula can be expressed as ordinal sum of Archimedean copulas and the minimum copula M we show that standard uniform convergence and convergence with respect to D are equivalent in Ca

Summary

Introduction

Various di erent notions of convergence in the family of bivariate copulas C have been considered in the literature: The standard uniform metric d∞ is probably the most common choice; since, d∞ is not capable of distinguishing independence and complete dependence (or, in the words of [24], d∞ does not ‘distinguish between di erent types of statistical dependence’) the stronger metric D was introduced in [36]. Sticking to Markov kernels/conditional distributions and considering weak convergence of almost all conditional distributions of copulas results in the notion of weak conditional convergence which was introduced recently in [17]. Weak conditional convergence implies convergence with respect to D , and it is straightforward to construct examples illustrating that the reverse implication is wrong (again see [17]). The question, whether weak conditional convergence is equivalent too, remains open - we have neither been able to prove equivalence nor to construct a counterexample.

Notation and preliminaries

The interrelation of Archimedean and associative copulas with respect to D

First suppose

Convergence in the class of associative copulas

Findings

Bδn it follows that