The Gaussian correlation inequality for multivariate centered Gaussian measures of centrally 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.

probability inequalities, multivariate normal distribution, multivariate gamma distribution, Gaussian correlation inequality, positive lower orthant dependence, monotonicity properties of multivariate gamma distributions.