Representation of the infimum and supremum of a family of multivariate distribution functions

Abstract

Sklar's theorem represents a single multivariate distribution function through its univariate marginal distributions and a copula. However, it fails in general when it comes to the representation of the envelopes of a family F of multivariate distribution functions (i.e. its point-wise infimum and supremum), even if we allow more general representing functions than copulas. In this paper we develop an alternative representation which describes the envelopes of a family F in terms of the copulas that correspond to the members of F. Our representation is reminiscent of Sklar's representation but is based on the idea of corner patches of copulas. We prove a series of four representation theorems; (1) a general one, which holds even if the envelopes of the family do not possess a Sklar type representation, (2) a version tailored for the case when the envelopes do possess a Sklar type representation, (3) a theorem for families of continuous distribution functions, and (4) a variant appropriate for modeling dependencies of absolutely continuous random variables. In addition, we demonstrate how our results can be applied to give a Sklar type theorem in the imprecise probability setting, which offers a new approach to multivariate probability boxes.