On a Generalized $*$-Product for Copulas

TL;DR

This paper generalizes the *-product for copulas to a new -product, addressing measurability issues and exploring its properties to better understand copula compatibility.

Contribution

It introduces a redefined -product for copulas, resolving measurability concerns and analyzing its properties for the first time.

Findings

Addressed measurability issues in the -product

Derived properties of the redefined -product

Enhanced understanding of copula compatibility

Abstract

This paper focuses on a generalization of the *-product called $\mathbf{C}$-product. This product, first introduced by Durante, Klement and Quesada-Molina, was used to characterize classes of compatible copulas. The $\mathbf{C}$-product of copulas $A$ and $B$ is defined to be an integral of a function which involves the copulas $A$ and $B$ and the family of copulas $\mathbf{C}$. However, measurability of the integrand in the definition is questionable. We will discuss this in details and attempt to re-define the product. Then we derive some properties of the re-defined product.