Affiliation:
1. Department of Mathematics and Statistics, McGill University , 805, rue Sherbrooke ouest , Montréal (Québec) , Canada H3A 0B9
Abstract
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.
Subject
Applied Mathematics,Modeling and Simulation,Statistics and Probability
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