On the distributional transform, Sklarʹs theorem, and the empirical copula process

We review the distributional transform of a random variable, some of its applications, and some related multivariate distributional transformations. The distributional transform is a useful tool, which allows in many respects to deal with general distributions in the same way as with continuous distributions. In particular it allows to give a simple proof of Sklarʹs theorem in the general case. It has been used in the literature for stochastic ordering results. It is also useful for an adequate definition of the conditional value at risk measure and for many further purposes. We also discuss the multivariate quantile transform as well as the multivariate extension of the distributional transform and some of their applications. In the final section we consider an application to an extension of a limit theorem for the empirical copula process, also called empirical dependence function, to general not necessarily continuous distributions. This is useful for constructing and analyzing tests of dependence properties for general distributions.