  The Gaussian correlation inequality (GCI) for symmetrical n-rectangles is
improved if the absolute components have a joint cumulative distribution (cdf)
which is MTP2 (multivariate totally positive of order 2). Inequalities of the
here given type hold at least for all MTP2-cdfs on R^n or (0,infinity)^n with
everywhere positive smooth densities. In particular, at least some infinitely
divisible multivariate chi-square distribution functions (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.
