论文标题
双变量高斯分布和三维高斯产品不等式的力矩比不平等
Moment ratio inequality of bivariate Gaussian distribution and three-dimensional Gaussian product inequality
论文作者
论文摘要
我们证明了三维高斯产品不平等（GPI）$ e [x_1^{2} x_2^{2m_2} x_3^{2m_3}] \ ge e [x_1^{2} {2 {2}] $（x_1，x_2，x_3）$和$ m_2，m_3 \ in \ mathbb {n} $。我们发现了矩比$ \ frac {| e [x_2^{2m_2+1} x_3^{2m_3+1}] |} {计算和硬分析之间的相互作用在证明中起着至关重要的作用。
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.
