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$R^n$andanyreal-valuedcontinuousnegativedefinitefunction$g: R^n\to R$theinequality$$Eg(X-Y)\le Eg(X+Y)$$holds\mathrm{.}Inparticular,for$a \in (0,2]$ and the Euclidean norm $\mathrm{\mid }\mathrm{.}\mathrm{\mid }$ one has $$E\mathrm{\mid }X-Y{\mathrm{\mid }}^a\le E\mathrm{\mid }X+Y{\mathrm{\mid }}^a\mathrm{.}$$ 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.