Sklar’s theorem by probabilistic continuation and two consistency results

We give a purely probabilistic proof of Sklar’s theorem by a simple continuing 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 from F : we construct a sequence of copula representers associated to the empirical distribution function of the sample which convergences a.s. to the representer of the copula function associated to F . Eventually, we extend the last theorem to the case where the marginals of F are discontinuous. This last result is discussed in relation to the inferential issues involved in copula models for discrete data.