Some New Results on Gaussian Product Inequalities

Abstract

The long-standing Gaussian product inequality (GPI) conjecture states that, for any centered ${\mathbb{R}}^n$-valued Gaussian random vector $(X_1,\dots ,X_n)$ and any positive reals ${\alpha }_1,\dots ,{\alpha }_n$, $\mathbf{E}[{\prod }_{j=1}^n|X_j{|}^{{\alpha }_j}]\ge {\prod }_{j=1}^n\mathbf{E}[|X_j{|}^{{\alpha }_j}]$. In this paper, we present some related inequalities for centered ${\mathbb{R}}^n$-valued Gaussian random vector $(X_1,\dots ,X_n)$ when $\{{\alpha }_1,\dots ,{\alpha }_n\}$ contains both positive and negative numbers.

15 pages