Regular Variation, Conditions of Domain of Attraction and the Existence of the Tail Dependence Function in the General Dependence Case: A Copula Approach

Abstract
Given a random vector X, Li and Sun (J Appl Probab 46:925–937, 2009), Weng and Zhang (J Multivar Anal 106:178–186, 2012) and Resnick (Extreme values. Regular variation and point processes, Springer, New York, 1987) proved relationships involving the vector being regularly varying, satisfying maximum domain of attraction conditions and the existence of its tail dependence function. Using the corresponding copula function, we give the conditions for the three properties to be equivalent in a general dependence framework. The main contribution of this work is to make well-known results on multivariate domain of attraction, regular variation and tail dependence interchangeable. We show how our results can be applied in a vector of empirical data with heterogeneous marginal distributions.