On convergence of associative copulas and related results
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 curren...
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De Gruyter
2021
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oai:doaj.org-article:6cd96376d1c14e20870b5a8a064bf2322021-12-05T14:10:46ZOn convergence of associative copulas and related results2300-229810.1515/demo-2021-0114https://doaj.org/article/6cd96376d1c14e20870b5a8a064bf2322021-10-01T00:00:00Zhttps://doi.org/10.1515/demo-2021-0114https://doaj.org/toc/2300-2298Triggered 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.Kasper Thimo M.Fuchs SebastianTrutschnig WolfgangDe Gruyterarticleassociative copulasarchimedean copulasweak convergencebaire category62h0560e0554e52Science (General)Q1-390MathematicsQA1-939ENDependence Modeling, Vol 9, Iss 1, Pp 307-326 (2021) |
| institution |
DOAJ |
| collection |
DOAJ |
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EN |
| topic |
associative copulas archimedean copulas weak convergence baire category 62h05 60e05 54e52 Science (General) Q1-390 Mathematics QA1-939 |
| spellingShingle |
associative copulas archimedean copulas weak convergence baire category 62h05 60e05 54e52 Science (General) Q1-390 Mathematics QA1-939 Kasper Thimo M. Fuchs Sebastian Trutschnig Wolfgang On convergence of associative copulas and related results |
| description |
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. |
| format |
article |
| author |
Kasper Thimo M. Fuchs Sebastian Trutschnig Wolfgang |
| author_facet |
Kasper Thimo M. Fuchs Sebastian Trutschnig Wolfgang |
| author_sort |
Kasper Thimo M. |
| title |
On convergence of associative copulas and related results |
| title_short |
On convergence of associative copulas and related results |
| title_full |
On convergence of associative copulas and related results |
| title_fullStr |
On convergence of associative copulas and related results |
| title_full_unstemmed |
On convergence of associative copulas and related results |
| title_sort |
on convergence of associative copulas and related results |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/6cd96376d1c14e20870b5a8a064bf232 |
| work_keys_str_mv |
AT kasperthimom onconvergenceofassociativecopulasandrelatedresults AT fuchssebastian onconvergenceofassociativecopulasandrelatedresults AT trutschnigwolfgang onconvergenceofassociativecopulasandrelatedresults |
| _version_ |
1718371728156000256 |