An Inequality Related to Negative Definite Functions

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 $$E g(X-Y) \le E g(X+Y)$$ 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.