Some New Results on Gaussian Product Inequalities

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 $\alpha_1, \dots, \alpha_n$, ${\bf E}[\prod_{j=1}^{n}|X_j|^{\alpha_j}]\ge \prod_{j=1}^{n}{\bf E}[|X_j|^{\alpha_j}]$. 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.