The Gaussian correlation inequality (GCI) for symmetrical n‑rectangles is improved if the absolute components have a joint cumulative distribution function (cdf), which is MTP₂ (multivariate totally positive of order 2). Inequalities of the given type hold at least for all MTP₂-cdfs on  or  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 broader class of multivariate gamma distributions are derived. 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 is also obtained. 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.

probability inequalities, Gaussian correlation inequality, multivariate chi-square distribution, multivariate gamma distribution, MTP₂-densities, infinitely divisible multivariate gamma distributions, M-matrices

Received: August 3, 2024; Revised: September 1, 2024; Accepted: September 24, 2024; Published: October 19, 2024

How to cite this article: Thomas Royen, On improved Gaussian correlation inequalities for symmetrical n-rectangles extended to certain multivariate gamma distributions and some further probability inequalities, Far East Journal of Theoretical Statistics 69(1) (2025), 1‑38. https://doi.org/10.17654/0972086325001

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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