Weakening the independence assumption on polar components: limit theorems for generalized elliptical distributions

Abstract

By considering the extreme behavior of bivariate random vectors with a polar representation R(u(T), v(T)), it is commonly assumed that the radial component R and the angular component T are stochastically independent. We investigate how to relax this rigid independence assumption such that conditional limit theorems can still be deduced. For this purpose, we introduce a novel measure for the dependence structure and present convenient criteria for validity of limit theorems possessing a geometrical meaning. Thus, our results verify a stability of the available limit results, which is essential in applications where the independence of the polar components is not necessarily present or exactly fulfilled.