On the Pickands stochastic process
Adja Mbarka Fall, Gane Samb Lo
Applied Mathematics, Gaston Berger University, Saint-Louis, Senegal

We introduce a Pickands process Pn(s) as a generalization of the classical Pickands estimate Pn(1/2) of the extremal index. We undertake here a purely stochastic process view for the asymptotic theory of that process by using the Csörgö-Csörgö-Horvàth-Mason weighted empirical and quantile process to suitable brownian bridge. This leads to its uniform convergence to the extremal index is studied as well as a weak convergence in the l([a,b]) to some Gaussian process {G(s), a ≤s ≤b} for all [a,b] in ]0,1. Greatly simplify the former results and enable applications based on stochastic processes methods.


Csörghö, M.,Csörghö, S., Horvàth, L. and Mason, M. (1986). Weighted empirical and quantile processes. Ann. Probab., 14, 31-85.

Drees. H. (1995). A refined Pickans Estimators for the extrem value index. Annals of Statistics Volume 23, Number 6, 2059-2080.

de Haan, L. and Feireira A. (2006). Extreme value theory: An introduction. Springer.

Lo, G. S.(1989). A note on the asymptotic normality of sums of extreme values. J. Statist. Plan. and Inf. 22, 89-94. (MR0996806)

Resnick, S.I. (1987). Extreme Values, Regular Variation and Point Processes. Springer-Verlag, New-York.

Shorack G.R. and Wellner J. A.(1986). Empirical Processes with Applications to Statistics. Wiley-Interscience, New-York.

Keywords: Stochastic process; Regularly varying; Brownian motion; Asymptotic normality

Biography: I'm Adja Mbarka Fall, born November 23, 1984 in Dakar, Senegal. I did all my university studies at the University Gaston Berger of Saint-Louis in the field of applied sciences and technologies. I am currently a PhD in Applied Statistics.