Convex rearrangements, generalized Lorenz curves, and correlated Gaussian data

Abstract

We propose a statistical index for measuring the fluctuations of a stochastic process ξ . This index is based on the generalized Lorenz curves and (modified) Gini indices of econometric theory. When ξ is a fractional Brownian motion with Hurst index 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 , 3 4 ) , α = 3 4 , or α ∈ ( 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.