The paper deals with the asymptotic laws of functional of standard random
variables. These classes of statistics are closely related to estimators of the
extreme value index when the underlying distribution function is in the Weibull
domain of attraction. We use techniques based on martingales theory to describe
the non Gaussian asymptotic distribution of the aforementioned statistics. We
provide results of a simulation study as well as statistical tests that may be
of interest with the proposed results.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1208.1487,
Submitted under the title : On The Weak Convergence Of The Functional Hill
Process For Small Parameters, Ti be published in Mathematical Methods of
Statistics 201
