Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

Convex rearrangements, generalized Lorenz curves, and correlated Gaussian data

Auteur(s):

Code(s) de Classification MSC:

• 60G18 Self-similar processes
• 60G15 Gaussian processes

Résumé: We propose a statistical index for measuring the fluctuations of a stochastic process $\xi$. This index is based on the generalized Lorenz curves and (modified) Gini indices of econometric theory. When $\xi$ is a fractional Brownian motion with Hurst index $\alpha\in(0,1)$, we develop a complete picture of the asymptotic theory of our index. In particular, we show that the asymptotic behaviour of our proposed index depends critically on whether $0<\alpha<3/4$, $\alpha=3/4$, or $3/4<\alpha<1$. Furthermore, in the first two cases, there is a Gaussian limit law, while the third case has an explicit limit law that is in the second Wiener chaos.

Mots Clés: Convex rearrangements ; Lorenz curves ; Gini indices ; fractional Brownian motion

Date: 2003-06-04

Prépublication numéro: PMA-825