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BY 4.0 license Open Access Published by De Gruyter Open Access October 18, 2021

On convergence of associative copulas and related results

  • Thimo M. Kasper , Sebastian Fuchs and Wolfgang Trutschnig EMAIL logo
From the journal Dependence Modeling


Triggered by a recent article establishing the surprising result that within the class of bivariate Archimedean copulas 𝒞ar different notions of convergence - standard uniform convergence, convergence with respect to the metric D1, 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 copula M we show that standard uniform convergence and convergence with respect to D1 are 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 with D1 a typical associative copula is Archimedean and a typical Archimedean copula is strict.

MSC 2010: 62H05; 60E05; 54E52


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Received: 2021-05-27
Accepted: 2021-07-22
Published Online: 2021-10-18

© 2021 Thimo M. Kasper et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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