Spatially homogeneous copulas

  • Fabrizio Durante
  • Juan Fernández Sánchez
  • Wolfgang Trutschnig


We consider spatially homogeneous copulas, i.e. copulas whose corresponding measure is invariant under a special transformations of \([0,1]^2\), and we study their main properties with a view to possible use in stochastic models. Specifically, we express any spatially homogeneous copula in terms of a probability measure on [0, 1) via the Markov kernel representation. Moreover, we prove some symmetry properties and demonstrate how spatially homogeneous copulas can be used in order to construct copulas with surprisingly singular properties. Finally, a generalization of spatially homogeneous copulas to the so-called (mn)-spatially homogeneous copulas is studied and a characterization of this new family of copulas in terms of the Markov \(*\)-product is established.


Copulas Dependence Probability measures Singular measures 



The authors gratefully acknowledge the support of the grant MTM2014-60594-P (partially supported by FEDER) from the Spanish Ministry of Economy and Competitiveness. The first author has been partially supported by INdAM–GNAMPA Project 2017 “Bounds for Risk Functionals in Dependence Models”.


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Copyright information

© The Institute of Statistical Mathematics, Tokyo 2018

Authors and Affiliations

  • Fabrizio Durante
    • 1
  • Juan Fernández Sánchez
    • 2
  • Wolfgang Trutschnig
    • 3
  1. 1.Dipartimento di Scienze dell’EconomiaUniversità del SalentoLecceItaly
  2. 2.Grupo de Investigación de Análisis MatemáticoUniversidad de AlmeríaAlmeríaSpain
  3. 3.Department for MathematicsUniversity of SalzburgSalzburgAustria

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