论文标题
使用平方和证明高斯产品不平等现象
Using Sums-of-Squares to Prove Gaussian Product Inequalities
论文作者
论文摘要
长期存在的高斯产品不平等（GPI）指出，$ e [\ prod_ {j = 1}^{n} x_j^{2m_j} {2m_j}] \ geq \ geq \ prod_ { $ m_1，\ dots，m_n \ in \ mathbb {n} $。在本文中，我们描述了一种计算算法，该算法涉及可用于解决GPI猜想的多元多项式总和的总和表示。要展示这种新颖方法的力量，我们将其应用于证明两个新的GPI：$ e [x_1^{2m_1} x_2^{6} x_3^{4} {4}] \ ge e [x_1^{2m_1}] $ e [x_1^{2m_1} x_2^{2} x_3^{2} x_4^{2}] \ ge e [x_1^{2m_1}] e [x_2^{2}]
The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}X_j^{2m_j}]\geq\prod_{j=1}^{n}E[X_j^{2m_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and $m_1,\dots,m_n\in\mathbb{N}$. In this paper, we describe a computational algorithm involving sums-of-squares representations of multivariate polynomials that can be used to resolve the GPI conjecture. To exhibit the power of this novel method, we apply it to prove two new GPIs: $E[X_1^{2m_1}X_2^{6}X_3^{4}]\ge E[X_1^{2m_1}]E[X_2^{6}]E[X_3^{4}]$ and $E[X_1^{2m_1}X_2^{2}X_3^{2}X_4^{2}]\ge E[X_1^{2m_1}]E[X_2^{2}]E[X_3^{2}]E[X_4^{2}]$.
