Probabilistic constructions of discrete copulas

For a multivariate vector X with discrete components, we construct, by means of explicit randomised transformations of X, multivariate couplings of copula representers U associated to X. As a result, we show that any copula can be constructed in this manner, giving a full probabilistic characterisation of the set of copula functions associated to X. The dependence properties of these copula representers are contrasted with those of the original X: the impact on independence and on the concordance order of this added randomisation structure is elucidated and quantified. Some explicit formulas are given in the bivariate and pattern recognition case. At last, an application of these constructions to the empirical copula shows they can easily be simulated when the distribution function is unknown.