Some probability inequalities for multivariate gamma and normal distributions

TL;DR

This paper extends probability inequalities from multivariate normal distributions to a broader class of multivariate gamma distributions, including non-integer degrees of freedom, and explores their monotonicity properties.

Contribution

It generalizes the Gaussian correlation inequality to smaller non-integer degrees of freedom and infinitely divisible multivariate gamma distributions, and proves a monotonicity property for increasing correlations.

Findings

Extended the inequality to smaller non-integer degrees of freedom.

Proved a monotonicity property for multivariate gamma distributions.

Established the inequality for infinitely divisible distributions.

Abstract

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.