Characterization of pre-idempotent Copulas

Chamnan Wongtawan () and Sumetkijakan Songkiat ()
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Chamnan Wongtawan: Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
Sumetkijakan Songkiat: Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Dependence Modeling, 2023, vol. 11, issue 1, 12

Abstract: Copulas C C for which ( C t C ) 2 = C t C {({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 L 2 {L}^{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 σ \sigma -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.

Keywords: idempotent copulas; pre-idempotent copulas; implicit dependence copulas; factorizable copulas; partial isometries (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:vrs:demode:v:11:y:2023:i:1:p:12:n:1

DOI: 10.1515/demo-2023-0106

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