Quantitative versions of the two-dimensional Gaussian product inequalities

Abstract

Abstract The Gaussian product-inequality (GPI) conjecture is one of the most famous inequalities associated with Gaussian distributions and has attracted much attention. In this note, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. For any centered, nondegenerate, and two-dimensional Gaussian random vector ( X 1 , X 2 ) $(X_{1}, X_{2})$ with E [ X 1 2 ] = E [ X 2 2 ] = 1 $E[X_{1}^{2}]=E[X_{2}^{2}]=1$ and the correlation coefficient ρ, we prove that for any real numbers α 1 , α 2 ∈ ( − 1 , 0 ) $\alpha _{1}, \alpha _{2}\in (-1,0)$ or α 1 , α 2 ∈ ( 0 , ∞ ) $\alpha _{1}, \alpha _{2}\in (0,\infty )$ , it holds that E [ | X 1 | α 1 | X 2 | α 2 ] − E [ | X 1 | α 1 ] E [ | X 2 | α 2 ] ≥ f ( α 1 , α 2 , ρ ) ≥ 0 , $$ {\mathbf{E}}\bigl[ \vert X_{1} \vert ^{\alpha _{1}} \vert X_{2} \vert ^{\alpha _{2}}\bigr]-{\mathbf{E}}\bigl[ \vert X_{1} \vert ^{ \alpha _{1}}\bigr]{\mathbf{E}}\bigl[ \vert X_{2} \vert ^{\alpha _{2}}\bigr]\ge f(\alpha _{1}, \alpha _{2}, \rho )\ge 0, $$ where the function f ( α 1 , α 2 , ρ ) $f(\alpha _{1}, \alpha _{2}, \rho )$ will be given explicitly by the Gamma function and is positive when ρ ≠ 0 $\rho \neq 0$ . When − 1 0 $\alpha _{2}>0$ , Russell and Sun (Statist. Probab. Lett. 191:109656, 2022) 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.

Keywords

• Gaussian product-inequality conjecture
• Quantitative inequality
• Hypergeometric function