Article ID Journal Published Year Pages File Type
5129495 Journal of Statistical Planning and Inference 2017 12 Pages PDF
Abstract

•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.

Related Topics
Physical Sciences and Engineering Mathematics Applied Mathematics
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