• Corpus ID: 88520179

A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions

@article{Royen2014ASP,
title={A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions},
author={Thomas Royen},
journal={arXiv: Probability},
year={2014}
}
• T. Royen
• Published 5 August 2014
• Mathematics
• arXiv: Probability
An extension of the Gaussian correlation conjecture (GCC) is proved for multivariate gamma distributions (in the sense of Krishnamoorthy and Parthasarathy). The classical GCC for Gaussian probability measures is obtained by the special case with one degree of freedom.
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References

SHOWING 1-10 OF 17 REFERENCES
The Gaussian Correlation Inequality for Symmetric Convex Sets
The paper is to prove the Gaussian correlation conjecture stating that, under the standard Gaussian measure, the measure of the intersection of any two symmetric convex sets is greater than or equal
A particular case of correlation inequality for the Gaussian measure
Our purpose is to prove a particular case of a conjecture concerning the Gaussian measure of the intersection of two symmetric convex sets of R n . This conjecture states that the measure of the
Rectangular Confidence Regions for the Means of Multivariate Normal Distributions
Abstract For rectangular confidence regions for the mean values of multivariate normal distributions the following conjecture of 0. J. Dunn [3], [4] is proved: Such a confidence region constructed
Infinite divisibility of multivariate gamma distributions and M - matrices
Suppose X=(X1,..,Xp)', has the Laplace transform ψ (t) = ∣Ι + VT∣-½, where V is a positive definite matrix and T= diag(t1,..,tp). It is shown that ψ(t) is infinitely divisible if and only if DV-1 D
A Geometric Approach to Radial Correlation Type Problems
A radial probability measure is a probability measure with a density (with respect to the Lebesgue measure) which depends only on the distances to the origin. Consider the Euclidean space enhanced
ON THE GAUSSIAN MEASURE OF THE INTERSECTION
• Mathematics
• 1998
The Gaussian correlation conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is
On the Gaussian measure of the intersection of symmetric, convex sets
• Mathematics
• 1996
The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is
Integral Representations and Approximations for Multivariate Gamma Distributions
AbstractLet R be a p×p-correlation matrix with an “m-factorial” inverse R−1 = D − BB′ with diagonal D minimizing the rank m of B. A new $$\left(m+1 \atop 2\right)$$-variate integral representation is
Inequalitites on the probability content of convex regions for elliptically contoured distributions
• History
• 1972
This work was supported in part by the National Science Foundation Grant No. 17172 at Stanford University, Grants No. 11021, 9593, and 21074 at the University of Minnesota and Grant No. 25911 at the