A multivariate Bahadur–Kiefer representation for the empirical copula process

We provide a multivariate extension of the Kiefer (1970) strong limit law for the uniform Bahadur–Kiefer reperesentation. This allows us to derive optimal rates for the strong approximation of empirical copula processes by sequences of Gaussian processes. We also provide a hill characterization of empirical copulas in a general framework. Bibliography: 30 titles.