Dependent Lindeberg CLT - Finite Dimensional for Empirical Processes of Cluster Functionals

TL;DR

This paper extends the CLT for empirical processes of cluster functionals from $eta$-mixing to a broader class of weakly dependent processes, demonstrating finite-dimensional convergence with practical examples and simulations.

Contribution

It generalizes the CLT for empirical processes of cluster functionals to weakly dependent processes, broadening applicability beyond $eta$-mixing.

Findings

Finite-dimensional CLT applies to weakly dependent processes.

Simulation confirms the practical effectiveness of the extended CLT.

Example with extremogram illustrates the theoretical results.

Abstract

Drees and Rootz\'en [2010] have proven central limit theorems (CLT) for empirical processes of extreme values cluster functionals built from $\beta$-mixing processes. The problem with this family of $\beta$-mixing processes is that it is quite restrictive, as has been shown by Andrews [1984]. We expand this result to a more general dependent processes family, known as weakly dependent processes in the sense of Doukhan and Louhichi [1999], but in finite-dimensional convergence (fidis). We show an example where the application of the CLT-fidis is sufficient in several cases, including a small simulation of the extremogram introduced by Davis and Mikosch [2009] to confirm the efficacy of our result.