Sklar's theorem for minitive belief functions

Abstract

In a former paper, the author has investigated how copulas can be used to express the dependence relation between two random sets. It has been proven that a joint belief function is related to its marginal belief functions by a family of copulas and that, in general, a single copula is not sufficient. In this paper the results are investigated under the assumption that the involved belief functions are minitive which corresponds to the important case where the associated random sets are consonant. It is proven that under this additional assumption a single copula is sufficient to express the dependence relation. In other words, this means that Sklar's theorem remains valid if joint and marginal distribution functions are replaced by joint and marginal minitive belief functions. • The dependence relation between random sets can be expressed by a family of copulas. • For consonant random sets one copula is sufficient. • The approach only makes use of the belief functions, no reference to specific random set representations is needed. • The relation to other approaches is discussed.