Some Smoothing Properties of the Star Product of Copulas

  • Wolfgang Trutschnig
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 190)


Three indications for the fact that the star product of copulas is smoothing are given. Firstly, it is shown that for every absolutely continuous copula A and every copula B both A*B and B*A are absolutely continuous. Secondly, an example of a singular copula A such that the absolutely continuous component of A*A has support [0,1]2 and mass at least 1/4 is given. Finally, it is shown that for every copula B of the form B = (1 − α)A + αS, whereby A is an absolutely continuous copula, S is a singular copula and α ∈ [0,1), there exists an absolutely continuous idempotent copula \(\widehat{B}\) such that \(\widehat{B}\) is the Cesáro limit of the sequence (B *n ) n ∈ ℕ of iterates of the star product of B with respect to the metric D 1 introduced in [15].


Copula star product 


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Research Unit for Intelligent Data AnalysisEuropean Centre for Soft ComputingMieresSpain

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