  The Gaussian correlation inequality for multivariate zero-mean normal
probabilities of symmetrical n-rectangles can be considered as an inequality
for multivariate gamma distributions (in the sense of Krishnamoorthy and
Parthasarathy [5]) with one degree of freedom. Its generalization to all
integer degrees of freedom and sufficiently large non-integer "degrees of
freedom" was recently proved in [10]. Here, this inequality is partly extended
to smaller non-integer degrees of freedom and in particular - in a weaker form
- to all infinitely divisible multivariate gamma distributions. A further
monotonicity property - sometimes called "more PLOD (positively lower orthant
dependent)" - for increasing correlations is proved for multivariate gamma
distributions with integer or sufficiently large degrees of freedom.
