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.
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).
"On weak conditional convergence of bivariate Archimedean and Extreme Value copulas, and consequences to nonparametric estimation." Bernoulli 27 (4) 2217 - 2240, November 2021. https://doi.org/10.3150/20-BEJ1306