A functional Generalized Hill process and applications

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 $f:\mathbb{N}\backslash 0⟼{\mathbb{R}}_{+}$ 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 $\mathcal{F}$.

23 pages

References in corpus (1)

• On the Generalized Hill Process for Small Parameters and Applications

Cited by in corpus (2)

• A supermartingale argument for characterizing the Functional Hill process weak law for small parameters
• Demmartingales and the functionnal Hill process for small parameters