Some smoothing properties of the star product of copulas

Abstract

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 B̂ such that B̂ is the Cesáro limit of the sequence (B*n)n∈ℕ of iterates of the star product of B with respect to the metric D1 introduced in [15]. © 2013 Springer-Verlag.