Empirical central limit theorem for cluster functionals without mixing

TL;DR

This paper establishes central limit theorems for empirical processes of extreme value cluster functionals without requiring mixing conditions, using coupling properties related to $ au$-dependence coefficients.

Contribution

It introduces new CLT results for cluster functionals under weaker dependence assumptions, especially for non-mixing processes like AR(1).

Findings

CLT proven for non-mixing AR(1) process.

Explicit covariance structure derived for the limit Gaussian process.

Simulations demonstrate the accuracy of extremal index estimation.

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.