A Functional Generalized Hill Process and Its Uniform Theory

Abstract

We are concerned in this paper with the functional asymptotic behavior of the sequence of stochastic processes
$$T_{n}(f)=\sum_{j=1}^{j=k}f(j)\left( \log X_{n-j+1,n}-\log X_{n-j,n}\right),\eqno(0.1)$$
indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N} \backslash \{0\} \longmapsto \mathbb{R}_{+}$ and where $k=k(n)$ satisfies
\begin{equation*}
1\leq k\leq n,k/n\rightarrow 0\text{ as }n\rightarrow \infty .
\end{equation*}
This is a functional generalized Hill process including as many new estimators of the extreme value index when $F$ is in the extreme value 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 $\mathcal{F}$.