Some new Gaussian product inequalities

Abstract

The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality E[X12X22m2X32m3]≥E[X12]E[X22m2]E[X32m3] for any centered Gaussian random vector (X1,X2,X3) and m2,m3∈N. First, we show that this inequality is implied by a combinatorial inequality. The combinatorial inequality can be verified directly for small values of m2 and arbitrary m3. Hence the corresponding cases of the three-dimensional inequality are proved. Second, we show that the three-dimensional inequality is equivalent to an improved Cauchy-Schwarz inequality. This observation leads us to derive some novel moment inequalities for bivariate Gaussian random variables.