Conditioning of copulas: Transformations, invariance and measures of concordance

In the present paper we study the problem of how to transform a copula for an arbitrary distribution function into a copula for its conditional distribution function where conditioning is meant with respect to a tail event in which the observations lie below some threshold. To this end, we consider conditioning of copulas as a map which transforms every copula into another one. Besides the general case, which refers to conditioning in all coordinates, we also pay attention to the special case of univariate conditioning, which refers to conditioning in a single coordinate. We investigate the behaviour of conditioning under composition and with respect to certain transformations of copulas, and we show that invariance of a copula under conditioning is equivalent to invariance of a copula under univariate conditioning in each coordinate. Finally, we apply conditioning of copulas to Sklar's Theorem and to measures of concordance.