Empirical central limit theorem for cluster functionals without mixing

Abstract

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.