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ON IMPROVED GAUSSIAN CORRELATION INEQUALITIES FOR SYMMETRICAL $n$-RECTANGLES EXTENDED TO CERTAIN MULTIVARIATE GAMMA DISTRIBUTIONS AND SOME FURTHER PROBABILITY INEQUALITIES
Keywords:
probability inequalities, Gaussian correlation inequality, multivariate chi-square distribution, multivariate gamma distribution, MTP2-densities, infinitely divisible multivariate gamma distributions, M-matrices.DOI:
https://doi.org/10.17654/0972086325001Abstract
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.
Received: August 3, 2024
Revised: September 1, 2024
Accepted: September 24, 2024
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