Empirical CLT for cluster functionals under weak dependence

Paul Doukhan, José Gregorio Gómez

Published 2015-07-22Version 1

We prove empirical central limit theorems (CLT) for extreme values cluster functionals empirical processes in the sense of the tough paper Drees and Rootz\'en (2010). Contrary to those authors we dont restrict to $\beta$ - mixing samples. For this we use coupling properties enlightened for Dedecker & Prieur's $\tau$ - dependence coefficients. We also explicit the asymptotic behavior of specific clusters. As an example we develop the number of excesses; it gives a complete example of a cluster functional for a non-mixing "reasonable model" (an AR(1)-process) for which results such as ours are definitely needed. In particular the expression of the limit Gaussian process is developed. 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.