Non-atomic bivariate copulas and implicitly dependent random variables

- Non-atomic bivariate copulas are defined in terms of two associated σ algebras.
- They are characterized in terms of its mass concentration and in terms of a partial factorizability of its Markov operators.
- Sufficient conditions for copulas with fractal support to be non-atomic and to be factorizable are given.

Two (continuous) random variables X and Y are implicitly dependent if there exist Borel functions α and β such that α∘X=β∘Y almost surely. The copulas of such random variables are exactly the copulas that are factorizable as the ∗-product of a left invertible copula and a right invertible copula. Consequently, every implicit dependence copula assigns full mass to 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 characterizations of a copula C assigning full mass to the graph of f(x)=g(y) in terms of a partial factorizability of its Markov operator TC and in terms of the non-atomicity of two newly defined associated σ-algebras σC and σC∗, in which case C is called non-atomic. As an application, we give a broad sufficient condition under which a copula with fractal support has an implicit dependence support. Under certain extra conditions, we explicitly compute the left invertible and right invertible factors of the copula with fractal support.