November 2021 On weak conditional convergence of bivariate Archimedean and Extreme Value copulas, and consequences to nonparametric estimation
Thimo M. Kasper, Sebastian Fuchs, Wolfgang Trutschnig
Author Affiliations +
Bernoulli 27(4): 2217-2240 (November 2021). DOI: 10.3150/20-BEJ1306


Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of convergence for copulas might seem far too restrictive to be of any practical importance – in fact, given samples of a copula C the corresponding empirical copulas do not converge weakly conditional to C with probability one in general. Within the class of Archimedean copulas and the class of Extreme Value copulas, however, standard pointwise convergence and weak conditional convergence can even be proved to be equivalent. Moreover, it can be shown that every copula C is the weak conditional limit of a sequence of checkerboard copulas. After proving these three main results and pointing out some consequences, we sketch some implications for two recently introduced dependence measures and for the nonparametric estimation of Archimedean and Extreme Value copulas.

Funding Statement

The first author gratefully acknowledges the financial support from Porsche Holding Austria and Land Salzburg within the WISS 2025 project ‘KFZ’ (P1900123). Moreover, the second and the third author gratefully acknowledge the support of the WISS 2025 project ‘IDA-lab Salzburg’ (20204-WISS/225/197-2019 and 0102-F1901166-KZP).


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Thimo M. Kasper. Sebastian Fuchs. Wolfgang Trutschnig. "On weak conditional convergence of bivariate Archimedean and Extreme Value copulas, and consequences to nonparametric estimation." Bernoulli 27 (4) 2217 - 2240, November 2021.


Received: 1 April 2020; Revised: 1 September 2020; Published: November 2021
First available in Project Euclid: 24 August 2021

MathSciNet: MR4303881
zbMATH: 1473.62160
Digital Object Identifier: 10.3150/20-BEJ1306

Keywords: Archimedean copula , checkerboard copula , dependence measure , estimation , extreme value copula , weak convergence

Rights: Copyright © 2021 ISI/BS


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Vol.27 • No. 4 • November 2021
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