# Preprint: 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:

• The RFBR; Grant 99-01-00112 (Y. D.)
• The National Science Foundation of the United States (D. Kh.)
• An NSERC of Canada individual research grant at the University of Western Ontario (R. Z.)
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 Youri.Davydov@univ-lille1.fr Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu Zhan Shi Laboratoire de Probabilités Université Paris VI 4 Place Jussieu F-75252 Paris, Cedex 05, France zhan@proba.jussieu.fr Ricardas Zitikis Department of Statistical and Actuarial Sciences University of Western Ontario London, Ontario, Canada, N6A 5B7 zitikis@stats.uwo.ca

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