Multivariate capacity functionals vs. capacity functionals on product spaces

Abstract

The aim of this paper is to study the relation between multivariate capacity functionals and capacity functionals on product spaces (also called joint capacity functionals). Multivariate capacity functionals characterize the joint distribution of finitely many random sets whereas joint capacity functionals characterize the distribution of a random set in a product space and as such provide an upper bound on a set of (joint) probability measures. It is demonstrated how to define a multivariate capacity functional from a given joint capacity functional, and how to extend a multivariate capacity functional to a joint capacity functional. In addition, it is shown how to construct a joint capacity functional from marginal maxitive capacity functionals when a copula is used for modelling stochastic dependence.