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
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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
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Zhan Shi
Laboratoire de Probabilités
Université Paris VI
4 Place Jussieu
F-75252 Paris, Cedex 05, France
zhan@proba.jussieu.fr
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Ricardas Zitikis
Department of Statistical and Actuarial Sciences
University of Western Ontario
London, Ontario, Canada, N6A 5B7
zitikis@stats.uwo.ca
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Last Update: May 29, 2003
© 2003 - Youri Davydov,
Davar Khoshnevisan, Zhan Shi, and Ricardas Zitikis