Convex rearrangements, generalized Lorenz curves, and correlated Gaussian data

We propose a statistical index for measuring the fluctuations of a stochastic process ξξ. This index is based on the generalized Lorenz curves and (modified) Gini indices of econometric theory.When ξξ is a fractional Brownian motion with Hurst index α∈(0,1)α∈(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 α∈(0,34), α=34, or α∈(34,1). 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.