On the gaussian measure of the intersection
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The Gaussian correlation conjecture states that for any two symmetric, convex sets in 2D 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 discussed by the authors.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 $c \sqrt{n}$. Further we show that if for every $n$ the conjecture is true when the sets are in the Euclidean ball of radius $\sqrt{n}$, then it is true in general. Our most concrete result is that the conjecture is true if the two sets are (arbitrary) centered ellipsoids.read more
Citations
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Book ChapterDOI
Gaussian processes: Inequalities, small ball probabilities and applications
Wenbo V. Li,Qi-Man Shao +1 more
TL;DR: In this paper, the authors focus on the inequalities, small-ball probabilities, and application of Gaussian processes, and find that the small ball probability is a key step in studying the lower limits of the Gaussian process.
Journal ArticleDOI
Approximation, metric entropy and small ball estimates for Gaussian measures
Wenbo V. Li,Werner Linde +1 more
TL;DR: This work relates the small ball behavior of a Gaussian measure μ on a Banach space E with the metric entropy behavior of K μ, the unit ball of the reproducing kernel Hilbert space of μ in E to enable the application of tools and results from functional analysis to small ball problems.
Book
Lectures on Gaussian Processes
TL;DR: The theory of Gaussian processes occupies one of the leading places in modern Probability as discussed by the authors, which is why Gaussian vectors and Gaussian distributions in infinite-dimensional spaces come into play.
Posted Content
A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions
TL;DR: An extension of the Gaussian correlation conjecture (GCC) for multivariate gamma distributions (in the sense of Krishnamoorthy and Parthasarathy) is proved in this article.
Book ChapterDOI
Royen’s Proof of the Gaussian Correlation Inequality
Rafał Latała,Dariusz Matlak +1 more
TL;DR: In this article, Royen's proof of the Gaussian correlation inequality was presented in detail, and it was shown that π(K ∩ L) ≥ μ(K)μ(L) for any centered Gaussian measure π on a symmetric convex set K, L in a convex space.
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).