An Opposite Gaussian Product Inequality

Abstract

The long-standing Gaussian product inequality (GPI) conjecture states that $E[{\prod }_{j=1}^n|X_j{|}^{{\alpha }_j}]\ge {\prod }_{j=1}^{n}E[|X_j{|}^{{\alpha }_j}]$ for any centered Gaussian random vector $(X_1,\dots ,X_n)$ and any non-negative real numbers ${\alpha }_j$, $j=1,\dots ,n$. 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.