Dependence Modeling (Apr 2022)

A topological proof of Sklar’s theorem in arbitrary dimensions

  • Benth Fred Espen,
  • Nunno Giulia Di,
  • Schroers Dennis

DOI
https://doi.org/10.1515/demo-2022-0103
Journal volume & issue
Vol. 10, no. 1
pp. 22 – 28

Abstract

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Copulas are appealing tools in multivariate probability theory and statistics. Nevertheless, the transfer of this concept to infinite dimensions entails some nontrivial topological and functional analytic issues, making a deeper theoretical understanding indispensable toward applications. In this short work, we transfer the well-known property of compactness of the set of copulas in finite dimensions to the infinite-dimensional framework. As an application, we prove Sklar’s theorem in infinite dimensions via a topological argument and the notion of inverse systems.

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