Convex Rearrangements, Generalized Lorenz Curves, and Correlated Gaussian Data

Youri Davydov, Davar Khoshnevisan, Zhan Shi, and Ricardas Zitikis

Abstract. 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 an asymptotic theory for our index. In particular, we show that the asymptotic behavior of our proposed index depends critically on whether \(\alpha\in(0\,,3/4)\), \(\alpha=3/4\), or \(\alpha\in(3/4\,,1)\). 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.

Keywords. Convex rearrangements, Lorenz curves, Gini indices, fractional Brownian motion

AMS Classification (2000) Primary. 60G18; Secondary. 60G15

Support. Research supported in part by grants from:

Pre/E-Prints. This paper is available as a prepublication of the University of Paris, prépublication PMA-825 (4/06/2003), in

Youri Davydov
Université des Sciences et Technologies de Lille
Laboratoire de Statistique et Probabilités
59655 Villeneuve
d'Ascq Cedex, France
Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.

Zhan Shi
Laboratoire de Probabilités
Université Paris VI
4 Place Jussieu
F-75252 Paris, Cedex 05, France
Ricardas Zitikis
Department of Statistical and Actuarial Sciences
University of Western Ontario
London, Ontario, Canada, N6A 5B7

Last Update: May 29, 2003
© 2003 - Youri Davydov, Davar Khoshnevisan, Zhan Shi, and Ricardas Zitikis