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

Abstract A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X = ( X 1 , … , X d ) {\boldsymbol{X}}=\left({X}_{1},\ldots ,{X}_{d}) of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on X {\boldsymbol{X}} is shown to be strictly weaker than the assumption that the density of the random vector ( ∣ X 1 ∣ , … , ∣ X d ∣ ) \left(| {X}_{1}| ,\ldots ,| {X}_{d}| ) is multivariate totally positive of order 2, abbreviated MTP 2 {\text{MTP}}_{2} , for which the GPI is already known to hold. Under this condition, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.