Supports of Implicit Dependence 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 $\ast$-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.

19 pages, 3 figures