Copula Measures and Sklar's Theorem in Arbitrary Dimensions

TL;DR

This paper introduces a unified framework for copulas as probability measures on general product spaces, extending Sklar's Theorem to infinite-dimensional settings and enabling modeling of dependencies in complex function spaces.

Contribution

It proposes a general definition of copulas for infinite-dimensional spaces and proves Sklar's Theorem in this broad context, unifying various existing concepts.

Findings

Proved Sklar's Theorem for infinite-dimensional copulas.

Established methods to transfer the theorem to function spaces.

Outlined approaches for modeling dependent measures in complex spaces.

Abstract

Although copulas are used and defined for various infinite-dimensional objects (e.g. Gaussian processes and Markov processes), there is no prevalent notion of a copula that unifies these concepts. We propose a unified approach and define copulas as probability measures on general product spaces. For this we prove Sklar's Theorem in this infinite-dimensional setting. We show how to transfer this result to various function space settings and describe how to model and approximate dependent probability measures in these spaces in the realm of copulas.