论文标题
相反的高斯产品不等式
An Opposite Gaussian Product Inequality
论文作者
论文摘要
长期存在的高斯产品不平等（GPI）猜想指出，$ e [\ prod_ {j = 1}^{n} | x_j |^{α_j}] \ geq \ geq \ geq \ prod_ { $（x_1，\ dots，x_n）$和任何非负实数$α_j$，$ j = 1，\ ldots，{n} $。在本说明中，当$ -1 <α_1<0 $和$α_2> 0 $：$ e [| x_1 |^α_1} | x_2 | x_2 |^{α_2} {α_2}] \ le Le时e [| x_1 |^{α_1}] e [| x_2 |^{α_2}] $。这完成了双变量高斯产品关系的图片。
The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}|X_j|^{α_j}]\geq\prod_{j=1}^{n}E[|X_j|^{α_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and any non-negative real numbers $α_j$, $j=1,\ldots,{n}$. In this note, we prove a novel "opposite GPI" for centered bivariate Gaussian random variables when $-1<α_1<0$ and $α_2>0$: $E[|X_1|^{α_1}|X_2|^{α_2}]\le E[|X_1|^{α_1}]E[|X_2|^{α_2}]$. This completes the picture of bivariate Gaussian product relations.
