Copulas in Hilbert spaces

TL;DR

This paper extends the concept of copulas to infinite-dimensional Hilbert spaces, establishing conditions for Sklar's theorem and constructing copulas via pairwise densities, advancing the theoretical framework in high-dimensional probability.

Contribution

It generalizes copulas to Hilbert spaces, proves a key part of Sklar's theorem in this setting, and provides a method to construct copulas using pairwise densities.

Findings

One direction of Sklar's theorem holds in Hilbert spaces.

A necessary and sufficient condition for the other direction is derived.

Constructed copulas in Hilbert spaces using pairwise copulas with densities.

Abstract

In this article, the concept of copulas is generalised to infinite dimensional Hilbert spaces. We show one direction of Sklar's theorem and explain that the other direction fails in infinite dimensional Hilbert spaces. We derive a necessary and sufficient condition which allows to state this direction of Sklar's theorem in Hilbert spaces. We consider copulas with densities and specifically construct copulas in a Hilbert space by a family of pairwise copulas with densities.