Weak convergence of the empirical copula process indexed by a class of functions is established. Two scenarios are considered in which either some smoothness of these functions or smoothness of the underlying copula function is required.
A novel integration by parts formula for multivariate, right-continuous functions of bounded variation, which is perhaps of independent interest, is proved. It is a key ingredient in proving weak convergence of a general empirical process indexed by functions of bounded variation.
"Weak convergence of empirical copula processes indexed by functions." Bernoulli 23 (4B) 3346 - 3384, November 2017. https://doi.org/10.3150/16-BEJ849