Some New Results on Gaussian Product Inequalities

TL;DR

This paper explores new inequalities related to the Gaussian product inequality conjecture, extending understanding to cases where the exponents include both positive and negative values for Gaussian vectors.

Contribution

It introduces novel inequalities for Gaussian vectors with mixed positive and negative exponents, advancing the theoretical framework of Gaussian product inequalities.

Findings

Derived inequalities for Gaussian vectors with mixed exponents

Extended the scope of GPI to include negative powers

Provided theoretical insights into Gaussian moment 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 $\alpha_1, \dots, \alpha_n$, ${\bf E}[\prod_{j=1}^{n}|X_j|^{\alpha_j}]\ge \prod_{j=1}^{n}{\bf E}[|X_j|^{\alpha_j}]$. 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.