A functional Generalized Hill process and applications

TL;DR

This paper investigates the asymptotic behavior of a generalized Hill process as a stochastic process, providing new insights into extremal index estimation and weak convergence in function spaces.

Contribution

It introduces a functional generalized Hill process with novel estimators for the extremal index and establishes its weak convergence properties.

Findings

Derived the functional asymptotic law of the process.

Provided explicit examples for specific classes of functions.

Extended the theory to include new extremal index estimators.

Abstract

We are concerned in this paper with the functional asymptotic behaviour of the sequence of stochastic processes T_{n}(f)=\sum_{j=1}^{j=k}f(j)(\log X_{n-j+1,n}-\log X_{n-j,n}), indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N} \backslash {0} \longmapsto \mathbb{R}_{+}$ and where $k=k(n)$ satisfies 1\leq k\leq n,k/n\rightarrow 0\text{as}n\rightarrow \infty. This is a functional generalized Hill process including as many new estimators of the extremal index when $F$ is in the extremal domain. We focus in this paper on its functional and uniform asymptotic law in the new setting of weak convergence in the space of bounded real functions. The results are next particularized for explicit examples of classes $\mathcal{F} $.