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Part of the book series: Advances in Intelligent Systems and Computing ((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].

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Trutschnig, W. (2013). Some Smoothing Properties of the Star Product of Copulas. In: Kruse, R., Berthold, M., Moewes, C., Gil, M., Grzegorzewski, P., Hryniewicz, O. (eds) Synergies of Soft Computing and Statistics for Intelligent Data Analysis. Advances in Intelligent Systems and Computing, vol 190. Springer, Berlin, Heidelberg.

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