Supports of Implicit Dependence Copulas

Abstract

A copula of continuous random variables X and Y is called an implicit dependence copula if there exist functions α and β such that α(X) = β(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 σalgebras. 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.