Some improved Gaussian correlation inequalities for symmetrical n-rectangles extended to some multivariate gamma distributions and some further probability inequalities
share
The Gaussian correlation inequality (GCI) for symmetrical n-rectangles is improved if the absolute components have a joint MTP2-distribution (multivariate totally positive of order 2). Inequalities of the here given type hold at least for all MTP2-probability measures on R^n or (0,infinity)^n with everywhere positive smooth densities. In particular, at least some infinitely divisible multivariate chi-square distributions (gamma distributions in the sense of Krishnamoorthy and Parthasarathy) with any positive real "degree of freedom" are shown to be MTP2. Moreover, further numerically calculable probability inequalities for a broad class of multivariate gamma distributions are derived and a different improvement for inequalities of the GCI-type - and of a similar type with three instead of two groups of components - with more special correlation structures. The main idea behind these inequalities is to find for a given correlation matrix with positive correlations a further correlation matrix with smaller correlations whose inverse is an M-matrix and where the corresponding multivariate gamma distribution function is numerically available.
READ FULL TEXT