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 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 (X1,X2) with E[X12]=E[X22]=1 and the correlation coefficient rho, we prove that for any real numbers alpha 1,alpha 2 is an element of(-1,0) or alpha 1,alpha 2 is an element of (0,infinity), it holds that E[|X1|alpha 1|X2|alpha 2]-E[|X1|alpha 1]E[|X2|alpha 2]>= f(alpha 1,alpha 2,rho)>= 0, where the function f(alpha 1,alpha 2,rho) will be given explicitly by the Gamma function and is positive when rho not equal 0. When -10, 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.

Automated summary

In this study, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. By employing the hypergeometric functions and the generalized hypergeometric functions, we prove that for any nondegenerate and centered two-dimensional Gaussian random vector (X1,X2) with a correlation coefficient rho, the inequality E[|X1|alpha1|X2|alpha2]-E[|X1|alpha1]E[|X2|alpha2]>= f(alpha1,alpha2,rho)>= 0 holds when alpha1 and alpha2 are either in (-1,0) or (0,infinity), and provide the explicit expression of the function f(alpha1,alpha2,rho). Furthermore, when -1<alpha1<0 and alpha2>0, we also prove the reverse Gaussian product inequality.