Characterization of pre-idempotent Copulas - Chamnan Wongtawan; Sumetkijakan Songkiat (Dependence Modeling, 2023)
Copulas CC for which (CtC)2=CtC{(C^{t}C)}^{2}=C^{t}C are called pre-idempotent copulas, of which well-studied examples are idempotent copulas and complete dependence copulas. As such, we shall work mainly with the topology induced by the modified Sobolev norm, with respect to which the class ℛ{\mathcal{ {\mathcal R} }} of pre-idempotent copulas is closed and the class of factorizable copulas is a dense subset of ℛ{\mathcal{ {\mathcal R} }}. Identifying copulas with Markov operators on L2L^{2}, the one-to-one correspondence between pre-idempotent copulas and partial isometries is one of our main tools. In the same spirit as Darsow and Olsen’s work on idempotent copulas, we obtain an explicit characterization of pre-idempotent copulas, which is split into cases according to the atomicity of its associated σσ -algebras, where the nonatomic case gives all factorizable copulas and the totally atomic case yields conjugates of ordinal sums of copies of the product copula.