An Inequality Related to Negative Definite Functions

Abstract

This is a substantially generalized version of the preprint arXiv:1105.4214 by Lifshits and Tyurin. We prove that for any pair of i.i.d. random vectors $X,Y$ in $R^n$ and any real-valued continuous negative definite function $g:R^n\to R$ the inequality $$Eg(X-Y)\le Eg(X+Y)$$ holds. In particular, for $a\in (0,2]$ and the Euclidean norm $|.|$ one has $$E|X-Y{|}^a\le E|X+Y{|}^a.$$ The latter inequality is due to A. Buja et al. (Ann. Statist., 1994} 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.