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On the Gaussian measure of the intersection of symmetric, convex sets
Gideon Schechtman,Joel Zinn +1 more
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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 greater than or equal to the product of the measures as mentioned in this paper.Abstract:
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 greater than or equal to the product of the measures In this paper we obtain several results which substantiate this conjecture For example, in the standard Gaussian case, we show there is a positive constant, c, such that the conjecture is true if the two sets are in the Euclidean ball of radius c √ n Further we show that if for every n the conjecture is true when the sets are in the Euclidean ball of radius √ n, then it is true in general Our most concrete result is that the conjecture is true if the two sets areread more
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Geometry of Isotropic Convex Bodies
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References
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Rectangular Confidence Regions for the Means of Multivariate Normal Distributions
TL;DR: For rectangular confidence regions for the mean values of multivariate normal distributions, this paper proved that a confidence region constructed for independent coordinates is, at the same time, a conservative confidence region for any case of dependent coordinates.
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On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
TL;DR: In this article, the authors extend the Prekopa-leindler theorem to other types of convex combinations of two positive functions and strengthen it by introducing the notion of essential addition.
Journal ArticleDOI
Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions
TL;DR: In this article, a function f(x) defined on X = X 1 × X 2 × × × X n where each X i is totally ordered satisfying f (x ∨ y) f(xi ∧ y) ≥ f(y) f (y), where the lattice operations ∨ and ∧ refer to the usual ordering on X, is said to be multivariate totally positive of order 2 (MTP2).