The Three-Dimensional Gaussian Product Inequality

Abstract

We prove the 3-dimensional Gaussian product inequality, i.e., for any real-valued centered Gaussian random vector $(X,Y,Z)$ and $m\in \mathbb{N}$, it holds that $\mathbf{E}[X^{2m}{Y}^{2m}{Z}^{2m}]\ge \mathbf{E}[X^{2m}]\mathbf{E}[Y^{2m}]\mathbf{E}[Z^{2m}]$. Our proof is based on some improved inequalities on multi-term products involving 2-dimensional Gaussian random vectors. The improved inequalities are derived using the Gaussian hypergeometric functions and have independent interest. As by-products, several new combinatorial identities and inequalities are obtained.