Dependent Lindeberg central limit theorem for the fidis of empirical processes of cluster functionals

Drees H. and Rootzen H. [Limit theorems for empirical processes of cluster functionals (EPCF). Ann Stat. 2010;38(4):2145-2186] have proven central limit theorems (CLTs) for EPCF built from beta-mixing processes. However, this family of beta-mixing processes is quite restrictive. We expand some of those results, for the finite-dimensional marginal distributions (fidis), to a more general dependent processes family, known as weakly dependent processes in the sense of Doukhan P. and Louhichi S. [A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 1999;84:313-342]. In this context, the CLT for the fidis of EPCF is sufficient in some applications. For instance, we prove the convergence without mixing conditions of the extremogram estimator, including a small example with simulation of the extremogram of a weakly dependent random process but nonmixing, in order to confirm the efficacy of our result.