An Opposite Gaussian Product Inequality

The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}|X_j|^{α_j}]\geq\prod_{j=1}^{n}E[|X_j|^{α_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and any non-negative real numbers $α_j$, $j=1,\ldots,{n}$. In this note, we prove a novel "opposite GPI" for centered bivariate Gaussian random variables when $-1<α_1<0$ and $α_2>0$: $E[|X_1|^{α_1}|X_2|^{α_2}]\le E[|X_1|^{α_1}]E[|X_2|^{α_2}]$. This completes the picture of bivariate Gaussian product relations.

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