  We prove central limit theorems (CLT) for empirical processes of extreme
values cluster functionals as in Drees and Rootz\'en (2010). We use coupling
properties enlightened for Dedecker \& Prieur's $\tau-$dependence coefficients
in order to improve the conditions of dependence and continue to obtain these
CLT. The assumptions are precisely set for particular processes and cluster
functionals of interest. The number of excesses provides a complete example of
a cluster functional for a simple non-mixing model (AR(1)-process) for which
ours results are definitely needed. We also give the expression explicit the
covariance structure of limit Gaussian process.
  Also we include in this paper some results of Drees (2011) for the extremal
index and some simulations for this index to demonstrate the accuracy of this
technique.
