On the Structure of the Classes of Copulas and Quasi-Copulas with a Given Diagonal Section

Abstract

In this paper we provide an alternative proof to that given in Ref. 1 to the fact that the class of multivariate copulas with a given diagonal section δ — denoted by 𝒞δ — is a singleton if, and only if, the unique element of the class is the copula of comonotonicity among random variables. This will be used to prove that, for any diagonal δ different from the diagonal of the comotonic copula, 𝒞δ is strictly included in the class of all multivariate quasi-copulas whose diagonal section is δ.