  The empirical copula process plays a central role for statistical inference
on copulas. Recently, Segers (2011) investigated the asymptotic behavior of
this process under non-restrictive smoothness assumptions for the case of
i.i.d. random variables. In the present paper we extend his main result to the
case of serial dependent random variables by means of the powerful and elegant
functional delta method. Moreover, we utilize the functional delta method in
order to obtain conditional consistency of certain bootstrap procedures.
Finally, we extend the results to the more general sequential empirical copula
process under serial dependence.
