Convex rearrangements, generalized Lorenz curves, and correlated Gaussian data

We propose a statistical index for measuring the fluctuations of a stochastic process. This index is based on the generalized Lorenz curves and (modified) Gini indices of econometric theory. When is a fractional Brownian motion with Hurst index alpha is an element of (0, 1), we develop a complete picture of the asymptotic theory of our index. In particular, we show that the asymptotic behavior of our proposed index depends critically on whether alpha is an element of (0, (3)/(4)), alpha = (3)/(4), or alpha is an element of (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. (c) 2006 Elsevier B.V. All rights reserved.