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=(X1,…,Xd){\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 (∣X1∣,…,∣Xd∣)\left(| {X}_{1}| ,\ldots ,| {X}_{d}| ) is multivariate totally positive of order 2, abbreviated MTP2{\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.

Keywords

• complete monotonicity
• gamma function
• gaussian product inequality
• gaussian random vector
• moment inequality
• multinomial
• multivariate normal
• polygamma function
• primary 60e15
• secondary 05a20
• 33b15
• 62e15
• 62h10
• 62h12