In multivariate extreme value theory dependence structures can be modeled by using Pickands dependence functions. Extreme value distribution functions (EVDs) with standard reversely exponential margins and the pertaining generalized Pareto distribution functions (GPDs) can be directly represented in terms of their Pickands dependence function D. Besides GPDs our statistical model comprises multivariate distribution functions belonging to the neighborhood of GPDs. They are characterized by two groups of density expansions which describe the dependence structure of the underlying random vectors and are the basis for the establishment of a test on tail dependence.
Because in important cases tail independence is attained at a very slow rate, the residual dependence structure plays a significant role. To analyze the residual dependence structure we deduce limiting distributions of maxima under triangular schemes of random vectors. Such a result has been investigated by Hüsler and Reiss and Hashorva in the special cases of normally and elliptically distributed random vectors respectively. Our aim is to treat the problem on an abstract level. For this purpose we study technical conditions imposed on the above mentioned density expansions and generalizations of the same conditions. We also extend our results to models with different univariate margins.
Finally, we present various measures of asymptotic dependence in the bivariate and multivariate framework. Analyses of these dependence measures within our statistical model show that they are related to certain density expansions and, in particular, to the Pickands dependence function.