Characterization of Multivariate Heavy-Tailed Distribution Families via Copula

The multivariate regular variation (MRV) is one of the most important tools in modelling multivariate heavy-tailed phenomena. This paper characterizes the MRV distribution through the upper tail dependence index of the copula associated with them. Along with Theorem 2.3 in Li and Sun (2009), our results imply that the existence of the upper tail dependence index of the copula is necessary and sufficient for a random vector with regularly varying univariate marginals to have a MRV tail. Moreover, the limit measure of the MRV tail can be explicitly characterized by the tail dependence index. Our analysis is also extended to two more general multivariate heavy-tailed distributions, including the Subexponential and the Long-tailed distribution families