Quantitative Versions of the Two-dimensional Gaussian Product Inequalities

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,\...