Mass distributions of two-dimensional extreme-value copulas and related results

Working with Markov kernels (conditional distributions) and right-hand derivatives D+A of Pickands dependence functions A we study the way two-dimensional extreme-value copulas (EVCs) CA distribute mass. Underlining the usefulness of working directly with D+A, we give first an alternative simple proof of the fact that EVCs with piecewise linear A can be expressed as weighted geometric mean of some EVCs whose dependence functions A have at most two edges and present a generalization of this result. After showing that the discrete component of the Markov kernel of CA concentrates its mass on the graphs of some increasing homeomorphisms ft, we determine which EVC assigns maximum mass to the union of the graphs of ft1,…,ftN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{t_{1}},\ldots ,f_{t_{N}}$\end{document}, derive the absolutely continuous component of an arbitrary EVC CA and deduce that the minimum copula M is the only (purely) singular EVC. Additionally, we prove the existence of EVCs CA which, despite their simple analytic form, exhibit the following surprisingly singular behavior: the discrete, the absolutely continuous and the singular component of the Lebesgue decomposition of the Markov kernel KCA(x,⋅)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K_{C_{A}}(x,\cdot )$\end{document} of CA have full support [0,1] for every x∈[0,1].