Copula Operad and Copula Entropy

Source abstract

We construct a symmetric operad $\mathfrak{C}$ on the class of all multivariate copulas, with operadic composition given by Sklar substitution. This places the hierarchical combination of dependence structures into an algebraic framework. Restricting to absolutely continuous copulas whose densities lie in $L\log L$, we prove that copula entropy is additive under composition, making it an additive character on suitable finite-entropy suboperads. We exhibit explicit closed suboperads (bounded, $L^{p}$, boundary-growth) and show by counterexample that componentwise $L\log L$ does not imply closure under composition.