Convex rearrangements of Gaussian processes

In this paper, we consider the asymptotic behavior of convex rearrangements for regularizations of paths of Gaussian processes with stationary increments, and we use the concentration principle to prove the almost sure convergence of these rearrangements to a nonrandom convex line, the so-called Lorentz curve, corresponding to the standard Gaussian law. Moreover, we obtain the same type of result for the Gaussian bridges of such processes. We also discuss relations with the recent results of Azais and Wschebor about the almost sure weak convergence of oscillations of Gaussian processes. As an application of our basic theorem we prove a theorem of Baxter type for $p$-variations of the paths and introduce a new family of consistent estimators of the fractal index.