Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**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*