Three-dimensional Gaussian product inequality with positive integer order moments

초록

The three-dimensional Gaussian product inequality conjecture states that for all positive real numbers p(1) , p(2) , and p(3) , and for all R-3-valued centered Gaussian random vectors (X-1 , X-2, X-3)(inverted perpendicular) with Var(X-i) > 0, i = 1, 2, 3, the inequality E [X-1|(p1) |X-2|(p2) |X-3|(p3)] >= E[X-1|(p1) ] E[X-2|(p2) ] E[X-3|(p3)] holds with equality if and only if X-1, X-2 and X-3 are independent. Recently, Herry, Malicet, and Poly (2024) showed that this conjecture is true when p(1), p(2), and p(3) are even positive integers. We extend this result to any positive integers p(1), p(2), and p(3). (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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