The three-dimensional Gaussian product inequality

We prove the 3-dimensional Gaussian product inequality, i.e., for any real-valued centered Gaussian random vector (X, Y, Z) and m is an element of N, it holds that E[X(2m)Y(2m)Z(2m)] >= E[X-2m]E[Y-2m]E[Z(2m)]. This settles positively the Gaussian product conjecture in the 3-dimensional case. Our proof is based on some improved inequalities on multi-term products involving 2-dimensional Gaussian random vectors. The improved inequalities are derived using the Gaussian hypergeometric functions and are of independent interest. As by-products, several new combinatorial identities and inequalities are obtained. (C) 2020 Elsevier Inc. All rights reserved.