Abstract
The three-dimensional Gaussian product inequality conjecture states that for all positive real numbers p1, p2, and p3, and for all R3-valued centered Gaussian random vectors (X1,X2,X3)⊤ with Var(Xi)>0, i=1,2,3, the inequality E[|X1|p1|X2|p2|X3|p3]≥E[|X1|p1]E[|X2|p2]E[|X3|p3] holds with equality if and only if X1,X2 and X3 are independent. Recently, Herry, Malicet, and Poly (2024) showed that this conjecture is true when p1, p2, and p3 are even positive integers. We extend this result to any positive integers p1, p2, and p3.
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