Sklar’s theorem derived using probabilistic continuation and two consistency results

We give a purely probabilistic proof of Sklar’s theorem by using a simple continuation technique and sequential arguments. We then consider the case where the distribution function FF is unknown but one observes instead a sample of i.i.d. copies distributed according to FF: we construct a sequence of copula representers associated with the empirical distribution function of the sample which convergences a.s. to the representer of the copula function associated with FF. Eventually, we are surprisingly able to extend the last theorem to the case where the marginals of FF are discontinuous.