Résumé
Many estimators of the extreme value index are functions of the $k_n$ largest observations of the sample and therefore can be seen as a functional of the $k_n$ upper tail quantile process. Under classical second order assumptions, this quantile process can be approximated, via a quantile transformation, by a non linear functional of the tail uniform empirical process. Here, we prove a strong invariance principle for this non linear functional. In some ways, this result improves the approximation result obtained by Drees since it is convenient to prove strong limit theorems. In particular, we obtain a functional law of the iterated logarithm for the quantile process. As an application, we establish a compact law of the iterated logarithm for the classical Hill estimator.