Some New Gaussian Product Inequalities

TL;DR

This paper investigates a specific Gaussian product inequality in three dimensions, linking it to combinatorial and improved Cauchy-Schwarz inequalities, and derives new moment inequalities for bivariate Gaussian variables.

Contribution

It establishes the three-dimensional Gaussian product inequality via combinatorial and Cauchy-Schwarz approaches, and introduces new moment inequalities for Gaussian variables.

Findings

The inequality is implied by a combinatorial inequality.

The inequality is equivalent to an improved Cauchy-Schwarz inequality.

New moment inequalities for bivariate Gaussian variables are derived.

Abstract

The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality $E[X_1^{2}X_2^{2m_2}X_3^{2m_3}]\ge E[X_1^{2}]E[X_2^{2m_2}]E[X_3^{2m_3}]$ for any centered Gaussian random vector $(X_1,X_2,X_3)$ and $m_2,m_3\in\mathbb{N}$. First, we show that this inequality is implied by a combinatorial inequality. The combinatorial inequality can be verified directly for small values of $m_2$ and arbitrary $m_3$. Hence the corresponding cases of the three-dimensional inequality are proved. Second, we show that the three-dimensional inequality is equivalent to an improved Cauchy-Schwarz inequality. This observation leads us to derive some novel moment inequalities for bivariate Gaussian random variables.