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.