On the Pickands stochastic process

We consider the Pickands process {equation*} P_{n}(s)=\log (1/s)^{-1}\log \frac{X_{n-k+1,n}-X_{n-[k/s]+1,n}}{% X_{n-[k/s]+1,n}-X_{n-[k/s^{2}]+1,n}}, {equation*} {equation*} (\frac{k}{n}\leq s^2 \leq 1), {equation*} which is a generalization of the classical Pickands estimate $P_n(1\mathrm{/}2)$ of the extremal index. We undertake here a purely stochastic process view for the asymptotic theory of that process by using the Cs"{o}rg\H{o}-Cs"{o}rg\H{o}-Horv'{a}th-Mason (1986) \cite{cchm} weighted approximation of the empirical and quantile processes to suitable Brownian bridges. This leads to the uniform convergence of the margins of this process to the extremal index and a complete theory of weak convergence of $P_n$ in ${\mathrm{\ell }}^{\mathrm{\infty }}([a,b])$ to some Gaussian process $$\{\mathbb{G},a\le s\le b\}$$ for all $[a,b]\subset ]0,1[$. This frame greatly simplifies the former results and enable applications based on stochastic processes methods.