Maximal coupling of empirical copulas for discrete vectors

For a vector X with a purely discrete multivariate distribution, we give simple short proofs of uniform a.s. convergence on their whole domain of two versions of genuine empirical copula functions, obtained either via probabilistic continuation, i.e. kernel smoothing, or via the distributional transform. These results give a positive answer to some delicate issues related to the convergence of copula functions in the discrete case. They are obtained under the very weak hypothesis of ergodicity of the sample, a framework which encompasses most types of serial dependence encountered in practice. Moreover, they allow to derive, as simple corollaries, almost sure consistency results for some recent extensions of concordance measures attached to discrete vectors. The proofs are based on a maximal coupling construction of the empirical cdf, a result of independent interest.