An Inequality Related to Negative Definite Functions

TL;DR

This paper proves a generalized inequality involving negative definite functions for i.i.d. random vectors, revealing new connections with bifractional Brownian motion and extending known results in multivariate statistics.

Contribution

It establishes a broad inequality for negative definite functions applied to i.i.d. vectors, generalizing previous results and linking to bifractional Brownian motion.

Findings

Proves that E g(X-Y) ≤ E g(X+Y) for negative definite g.

Shows the inequality for Euclidean norm powers, extending prior work.

Identifies a connection with bifractional Brownian motion and provides counter-examples.

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