# The Three-Dimensional Gaussian Product Inequality

## TL;DR

This paper proves the 3-dimensional Gaussian product inequality, establishing a lower bound for the expectation of the product of powers of Gaussian variables, using advanced inequalities involving hypergeometric functions.

## Contribution

It introduces a novel proof of the 3D Gaussian product inequality using improved inequalities with hypergeometric functions, and derives new combinatorial identities.

## Key findings

- Proved the 3D Gaussian product inequality.
- Developed improved inequalities for Gaussian vectors.
- Derived new combinatorial identities.

## 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 ${\mathbf{E}}[X^{2m}Y^{2m}Z^{2m}]\geq{\mathbf{E}}[X^{2m}]{\mathbf{E}}[Y^{2m}]{\mathbf{E}}[Z^{2m}]$. 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.