  Drees and Rootz\'en [2010] have proven central limit theorems (CLT) for
empirical processes of extreme values cluster functionals built from
$\beta$-mixing processes. The problem with this family of $\beta$-mixing
processes is that it is quite restrictive, as has been shown by Andrews [1984].
We expand this result to a more general dependent processes family, known as
weakly dependent processes in the sense of Doukhan and Louhichi [1999], but in
finite-dimensional convergence (fidis). We show an example where the
application of the CLT-fidis is sufficient in several cases, including a small
simulation of the extremogram introduced by Davis and Mikosch [2009] to confirm
the efficacy of our result.
