Moment ratio inequality of bivariate Gaussian distribution and three-dimensional Gaussian product inequality

Oliver Russell; Wei Sun

paper

Moment ratio inequality of bivariate Gaussian distribution and three-dimensional Gaussian product inequality

arXiv:2208.13957

Abstract

We prove the three-dimensional Gaussian product inequality (GPI) $E [X_{1}^{2} X_{2}^{2 m_{2}} X_{3}^{2 m_{3}}] \geq E [X_{1}^{2}] E [X_{2}^{2 m_{2}}] E [X_{3}^{2 m_{3}}]$ for any centered Gaussian random vector $(X_{1}, X_{2}, X_{3})$ and $m_{2}, m_{3} \in N$ . We discover a novel inequality for the moment ratio $\frac{∣ E [ X _{2}^{2 m_{2} + 1} X _{3}^{2 m_{3} + 1} ] ∣}{E [ X _{2}^{2 m_{2}} X _{3}^{2 m_{3}} ]}$ , which implies the 3D-GPI. The interplay between computing and hard analysis plays a crucial role in the proofs.