论文标题
二维高斯产品不平等的定量版本
Quantitative Versions of the Two-dimensional Gaussian Product Inequalities
论文作者
论文摘要
高斯产品不平等（GPI）猜想是与高斯分布相关的最著名的不平等之一，引起了很多关注。在本说明中，我们研究了二维高斯产品不平等的定量版本。对于任何中心的非分二维高斯随机矢量$（x_1，x_2）$，带有差异$σ_1^2，σ_2^2 $和相关系数$ $ρ$，我们证明，对于任何实数$α_1，α_2该％存在$α_1，α_2$和$ρ$的功能，使得$$ {\ bf e} [| x_1 |^{α_1} | x_2 | x_2 |^{α_2}] - {\ bf e} e} [| x_2 |^{α_2}] \ ge f（σ_1，σ_2，α_1，α_1，α_2，α_2，ρ）\ ge 0，$$，其中函数$ f（σ_1，σ_2，σ_2，α_1，α_1，α_2，ρ）$将通过gam ne $ ne $ ne $ ne $ ne $ ne $ ne $ ne $ ne $ ne $ ne $ ne $ ne $ ne $。当$ -1 <α_1<0 $和$α_2>0时，$ Russell和Sun（Arxiv：2205.10231V1）证明了“对立的高斯产品不平等”，我们还将提供定量版本。这些定量不平等是通过采用高几何函数和广义高几幅功能来得出的。
The Gaussian product inequality (GPI) conjecture is one of the most famous inequalities associated with Gaussian distributions and has attracted a lot of concerns. In this note, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. For any centered non-degenerate two-dimensional Gaussian random vector $(X_1, X_2)$ with variances $σ_1^2, σ_2^2$ and the correlation coefficient $ρ$, we prove that for any real numbers $α_1, α_2\in (-1,0)$ or $α_1, α_2\in (0,\infty)$, it holds that %there exist functions of $α_1, α_2$ and $ρ$ such that $${\bf E}[|X_1|^{α_1}|X_2|^{α_2}]-{\bf E}[|X_1|^{α_1}]{\bf E}[|X_2|^{α_2}]\ge f(σ_1,σ_2,α_1, α_2, ρ)\ge 0, $$ where the function $f(σ_1,σ_2,α_1, α_2, ρ)$ will be given explicitly by Gamma function and is positive when $ρ\neq 0$. When $-1<α_1<0$ and $α_2>0,$ Russell and Sun (arXiv: 2205.10231v1) proved the "opposite Gaussian product inequality", of which we will also give a quantitative version. These quantitative inequalities are derived by employing the hypergeometric functions and the generalized hypergeometric functions.
