A combinatorial proof of the Gaussian product inequality conjecture beyond the MTP2 case

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 $\boldsymbol{X} = (X_1,X_2,\dots,X_d)$ 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 $|\boldsymbol{X}| = (|X_1|,|X_2|,\dots,|X_d|)$ to be in the multivariate totally positive of order $2$ ($\mathrm{MTP}_2$) class on $[0,\infty)^d$, for which the conjecture is already known to be true.