Product of copulas

A few days ago I wrote a post about copulas and operations on them that have a group structure. Here's another example of group structure for copulas. As in the previous post I'm just looking at two-dimensional copulas to keep things simple.

Given two copulas C₁ and C₂, you can define a sort of product between them by

Here D_i is the partial derivative with respect to the ith variable.

The product of two copulas is another copula. This product is associative but not commutative. There is an identity element, so copulas with this product form a semigroup.

The identity element is the copula

that is,

for any copula C.

The copula M is important because it is the upper bound for the Fri(C)chet-Hoeffding bounds: For any copula C,

There is also a sort of null element for our semigroup, and that is the independence copula

It's called the independence copula because it's the copula for two independent random variables: their joint CDF is the product of their individual CDFs. It acts like a null element because

This tells us we have a semigroup and not a group: the independence copula cannot have an inverse.

Reference: Roger B. Nelsen. An Introduction to Copulas.