An Opposite Gaussian Product Inequality

TL;DR

This paper proves a new inequality for bivariate Gaussian variables with specific negative and positive exponents, complementing the classical Gaussian product inequality and completing the understanding of such relations.

Contribution

It introduces an 'opposite GPI' for certain bivariate Gaussian variables, expanding the scope of Gaussian product inequalities.

Findings

Established the opposite GPI for specific exponents

Completed the characterization of bivariate Gaussian product relations

Provides new insights into Gaussian inequalities

Abstract

The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}|X_j|^{\alpha_j}]\geq\prod_{j=1}^{n}E[|X_j|^{\alpha_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and any non-negative real numbers $\alpha_j$, $j=1,\ldots,{n}$. In this note, we prove a novel "opposite GPI" for centered bivariate Gaussian random variables when $-1<\alpha_1<0$ and $\alpha_2>0$: $E[|X_1|^{\alpha_1}|X_2|^{\alpha_2}]\le E[|X_1|^{\alpha_1}]E[|X_2|^{\alpha_2}]$. This completes the picture of bivariate Gaussian product relations.