A Probabilistic Inequality Related to Negative Definite Functions

Abstract

We prove that for any pair of i.i.d. random vectors X,Y in $\mathbb{R}^n$ and any real-valued continuous negative definite function $\psi\; : \;\mathbb{R}^n\rightarrow\mathbb{R}$ the inequality

$$\mathbb{E}\;\psi\;(X\;-\;Y)\leqslant\mathbb{E}\;\psi\;(X\;+\;Y).$$

holds. In particular, for $\alpha\;\in\;(0,2]$ and the Euclidean norm $\|\cdot\|_2$ one has

$$\mathbb{E}\|(X\;-\;Y)\|^\alpha_2\leqslant\mathbb{E}\|(X\;+\;Y)\|^\alpha_2.$$

The latter inequality is due to A. Buja et al. [4] where it is used for some applications in multivariate statistics. We show a surprising connection with bifractional Brownian motion and provide some related counter-examples.

Mathematics Subject Classification (2010). Primary 60E15; Secondary 60G22, 60E10.