Tail Dependence of Multivariate Archimedean Copulas

Abstract: Archimedean copulas generated by Laplace transforms have been extensively studied in the literature, with much of the focus on tail dependence limited only to cases where the Laplace transforms exhibit regular variation with positive tail indices. In this paper, we extend the investigation to include Archimedean copulas associated with both slowly varying and rapidly varying Laplace transforms. We show that tail dependence functions with various tail orders effectively capture the extremal dependence across the entire class of Archimedean copulas, reflecting the full spectrum of tail behaviors exhibited by the underlying Laplace transforms.