An Empirical Central Limit Theorem with applications to copulas under weak dependence

We state a multidimensional Functional Central Limit Theorem for weakly dependent random vectors. We apply this result to copulas. We get the weak convergence of the empirical copula process and of its smoothed version. The finite dimensional convergence of smoothed copula densities is also proved. A new definition and the theoretical analysis of conditional copulas and their empirical counterparts are provided.