An Inequality Related to Bifractional Brownian Motion

TL;DR

This paper establishes a novel inequality involving i.i.d. random variables and connects it to bifractional Brownian motion, extending the result to Bernstein functions and providing counterexamples.

Contribution

It introduces a new inequality for i.i.d. variables and links it to bifractional Brownian motion, extending the scope with Bernstein functions and counterexamples.

Findings

Proved that E|X-Y|^a ≤ E|X+Y|^a for i.i.d. variables with finite a-th moment.

Connected the inequality to bifractional Brownian motion.

Extended the inequality to Bernstein functions and provided counterexamples.

Abstract

We prove that for any pair of i.i.d. random variables $X,Y$ with finite moment of order $a \in (0,2]$ it is true that $E |X-Y|^a \leq E |X+Y|^a$. Surprisingly, this inequality turns out to be related with bifractional Brownian motion. We extend this result to Bernstein functions and provide some counter-examples.