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

We prove the three-dimensional Gaussian product inequality (GPI) E[X12X22m2X2m3 3] >= E[X12]E[X22m2]E[X2m3 3] for any centered Gaussian random vector (X1, X2, X3) and m2, m3 is an element of N. We discover a novel inequality for the moment ratio |E[X 2m2+1X2m3+1 2 3]| E[X22m2X2m3 3] , which implies the 3D-GPI. The interplay between computing and hard analysis plays a crucial role in the proofs. (c) 2023 Elsevier Inc. All rights reserved.