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 F is unknown but one observes instead a sample of i.i.d. copies distributed according to F : 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 F . Eventually, we are surprisingly able to extend the last theorem to the case where the marginals of F are discontinuous.