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.

中文翻译：

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关于给定对角线截面的 Copulas 和 Quasi-Copula 类的结构

在本文中，我们提供了参考文献中给出的替代证明。1 到这样一个事实，即具有给定对角线截面 δ 的多元 copula 类 — 用 𝒞 表示δ— 是单例当且仅当该类的唯一元素是随机变量之间的单调性的 copula。这将用于证明，对于任何不同于同调系对角线的对角线 δ，𝒞δ严格包含在对角线截面为 δ 的所有多元拟联结类中。