论文标题
一些新的高斯产品不平等
Some New Gaussian Product Inequalities
论文作者
论文摘要
高斯产品不平等是一个长期的猜想。在本文中，我们研究了三维不等式$ e [x_1^{2} x_2^{2m_2} {2m_2} x_3^{2m_3} {2m_3}] \ ge e [x_1^{2} {2}] e [x_2^{2m_2}] $（x_1，x_2，x_3）$和$ m_2，m_3 \ in \ mathbb {n} $。首先，我们表明这种不平等是组合不平等所暗示的。可以直接以$ M_2 $和任意$ M_3 $的小值直接验证组合不平等。因此，证明了三维不平等的相应案例。其次，我们表明三维不平等等同于改善的Cauchy-Schwarz不平等。这一观察结果使我们得出了一些新型的时刻不平等，对双变量高斯随机变量。
The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality $E[X_1^{2}X_2^{2m_2}X_3^{2m_3}]\ge E[X_1^{2}]E[X_2^{2m_2}]E[X_3^{2m_3}]$ for any centered Gaussian random vector $(X_1,X_2,X_3)$ and $m_2,m_3\in\mathbb{N}$. First, we show that this inequality is implied by a combinatorial inequality. The combinatorial inequality can be verified directly for small values of $m_2$ and arbitrary $m_3$. Hence the corresponding cases of the three-dimensional inequality are proved. Second, we show that the three-dimensional inequality is equivalent to an improved Cauchy-Schwarz inequality. This observation leads us to derive some novel moment inequalities for bivariate Gaussian random variables.
