A combinatorial proof of the Gaussian product inequality conjecture beyond the MTP2 case
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In this paper, we present a combinatorial proof of the Gaussian product inequality (GPI) conjecture in all dimensions when the components of the centered Gaussian vector X = (X_1,X_2,…,X_d) can be written as linear combinations, with nonnegative coefficients, of the components of a standard Gaussian vector. The proof comes down to the monotonicity of a certain ratio of gamma functions. We also show that our condition is weaker than assuming the vector of absolute values |X| = (|X_1|,|X_2|,…,|X_d|) to be in the multivariate totally positive of order 2 (MTP_2) class on [0,∞)^d, for which the conjecture is already known to be true.
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