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Moment ratio inequality of bivariate Gaussian distribution and three-dimensional Gaussian product inequality
Oliver Russell, Wei Sun · 2022-08-30 · via math.PR updates on arXiv.org

We prove the three-dimensional Gaussian product inequality (GPI) $E[X_1^{2}X_2^{2m_2}X_3^{2m_3}]\ge E[X_1^{2}]E[X_2^{2m_2}]E[X_3^{2m_3}]$ for any centered Gaussian random vector $(X_1,X_2,X_3)$ and $m_2,m_3\in\mathbb{N}$. We discover a novel inequality for the moment ratio $\frac{|E[ X_2^{2m_2+1}X_3^{2m_3+1}]|}{E[ X_2^{2m_2}X_3^{2m_3}]}$, which implies the 3D-GPI. The interplay between computing and hard analysis plays a crucial role in the proofs.

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