Levy Measures of Infinitely Divisible Random Vectors and Slepian Inequalities

Abstract

We study Slepian inequalities for general non-Gaussian infinitely divisible random vectors. Conditions for such inequalities are expressed in terms of the corresponding Levy measures of these vectors. These conditions are shown to be nearly best possible, and for a large subfamily of infinitely divisible random vectors these conditions are necessary and sufficient for Slepian inequalities. As an application we consider symmetric $\alpha$-stable Ornstein-Uhlenbeck processes and a family of infinitely divisible random vectors introduced by Brown and Rinott.