Quantitative Versions of the Two-dimensional Gaussian Product Inequalities

Abstract

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 $\ddot{I}{ƒ}_1^2,\ddot{I}{ƒ}_2^2$ and the correlation coefficient $\ddot{I}$, we prove that for any real numbers $\hat{I}{\pm }_1,\hat{I}{\pm }_2\in (-1,0)$ or $\hat{I}{\pm }_1,\hat{I}{\pm }_2\in (0,\infty )$, it holds that %there exist functions of $\hat{I}{\pm }_1,\hat{I}{\pm }_2$ and $\ddot{I}$ such that $$\mathbf{E}[|X_1{|}^{\hat{I}{\pm }_1}|X_2{|}^{\hat{I}{\pm }_2}]-\mathbf{E}[|X_1{|}^{\hat{I}{\pm }_1}]\mathbf{E}[|X_2{|}^{\hat{I}{\pm }_2}]\ge f(\ddot{I}{ƒ}_1,\ddot{I}{ƒ}_2,\hat{I}{\pm }_1,\hat{I}{\pm }_2,\ddot{I})\ge 0,$$ where the function $f(\ddot{I}{ƒ}_1,\ddot{I}{ƒ}_2,\hat{I}{\pm }_1,\hat{I}{\pm }_2,\ddot{I})$ will be given explicitly by Gamma function and is positive when $\ddot{I}\ne 0$. When $-1<\hat{I}{\pm }_1<0$ and $\hat{I}{\pm }_2>0,$ 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.

10 pages