Some improved Gaussian correlation inequalities for symmetrical n-rectangles extended to some multivariate gamma distributions and some further probability inequalities

Abstract

The Gaussian correlation inequality (GCI) for symmetrical n-rectangles is improved if the absolute components have a joint MTP 2 -distribution (multivariate totally positive of order 2). Inequalities of the here given type hold at least for all MTP 2 -probability measures on R n or R 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 MTP 2 . Moreover, further numerically calculable probability inequalities for a broad class of multivariate gamma distributions are derived and a different improvement for inequalities of the GCItype -and of a similar type with three instead of two groups of componentswith 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.

Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

“…For the aforementioned multivariate gamma distribution (diagonal blocks of width 1), this inequality, formerly known as the Gaussian correlation inequality conjecture, was settled by Thomas Royen in 2014, see Royen [25] and also Royen [26,27] for some extensions. It is therefore natural to conjecture that the same inequality holds more broadly for diagonal blocks of any widths p 1 , .…”