On the distribution of Pickands coordinates in bivariate EV and GP models

Let (U,V) be a random vector with U[less-than-or-equals, slant]0, V[less-than-or-equals, slant]0. The random variables Z=V/(U+V), C=U+V are the Pickands coordinates of (U,V). They are a useful tool for the investigation of the tail behavior in bivariate peaks-over-threshold models in extreme value theory. We compute the distribution of (Z,C) among others under the assumption that the distribution function H of (U,V) is in a smooth neighborhood of a generalized Pareto distribution (GP) with uniform marginals. It turns out that if H is a GP, then Z and C are independent, conditional on C>c[greater-or-equal, slanted]-1. These results are used to derive approximations of the empirical point process of the exceedances (Zi,Ci) with Ci>c in an iid sample of size n. Local asymptotic normality is established for the approximating point process in a parametric model, where c=c(n)[short up arrow]0 as n-->[infinity].