Toward a Copula Theory for Multivariate Regular Variation

Multivariate regular variation describes the relative decay rates of joint tail probabilities of a random vector with respect to tail probabilities of a norm (any norm) of this random vector, and it is often used in studying heavy-tail phenomena observed in data analysis in various fields, such as finance and insurance. Multivariate regular variation can be analyzed in terms of the intensity measure or spectral measure, but can also be studied by using the copula approach. In this paper, the basic ingredients of a measure-theoretic copula theory for multivariate regular variation are presented, and the method is based on extraction of scale-invariant extremal dependence from the intensity measure by standardizing its margins. Various examples as well as the advantages and disadvantages of the copula approach are also discussed.