Supports of Implicit Dependence Copulas

TL;DR

This paper characterizes copulas supported on graphs of measure-preserving functions, focusing on implicit dependence copulas, and provides conditions and explicit factorizations for self-similar copulas.

Contribution

It offers a new characterization of implicit dependence copulas via associated sigma-algebras and analyzes their structure, especially for self-similar copulas.

Findings

Characterization of implicit dependence supports using non-atomicity of sigma-algebras

Broad sufficient condition for self-similar copulas to have implicit dependence supports

Explicit computation of factors for certain self-similar copulas

Abstract

A copula of continuous random variables $X$ and $Y$ is called an \emph{implicit dependence copula} if there exist functions $\alpha$ and $\beta$ such that $\alpha(X) = \beta(Y)$ almost surely, which is equivalent to $C$ being factorizable as the $*$-product of a left invertible copula and a right invertible copula. Every implicit dependence copula is supported on the graph of $f(x) = g(y)$ for some measure-preserving functions $f$ and $g$ but the converse is not true in general. We obtain a characterization of copulas with implicit dependence supports in terms of the non-atomicity of two newly defined associated $\sigma$-algebras. As an application, we give a broad sufficient condition under which a self-similar copula has an implicit dependence support. Under certain extra conditions, we explicitly compute the left invertible and right invertible factors of the self-similar copula.