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Displaying similar documents to “Shift inequalities of Gaussian type and norms of barycentres”

Geometric influences II: Correlation inequalities and noise sensitivity

Nathan Keller, Elchanan Mossel, Arnab Sen (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand’s lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini–Kalai–Schramm (BKS) noise sensitivity...

The Gaussian zoo.

Renze, John, Wagon, Stan, Wick, Brian (2001)

Experimental Mathematics

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On Gaussian Brunn-Minkowski inequalities

Franck Barthe, Nolwen Huet (2009)

Studia Mathematica

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We are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrhard inequality for m Borel or convex sets based on a previous work by Borell. Our method also yields semigroup proofs of the geometric Brascamp-Lieb inequality and of its reverse form, which follow exactly the same lines.

Gaussian Integers

Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, Yasunari Shidama (2013)

Formalized Mathematics

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Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic. ...

Robust optimality of Gaussian noise stability

Elchanan Mossel, Joe Neeman (2015)

Journal of the European Mathematical Society

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We prove that under the Gaussian measure, half-spaces are uniquely the most noise stable sets. We also prove a quantitative version of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This extends a theorem of Borell, who proved the same result but without uniqueness, and it also answers a question of Ledoux, who asked whether it was possible to prove Borell’s theorem using a direct semigroup argument. Our quantitative uniqueness result...

SURE shrinkage of gaussian paths and signal identification

Nicolas Privault, Anthony Réveillac (2011)

ESAIM: Probability and Statistics

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Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes, based on their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we derive an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise.

On small deviations of Gaussian processes using majorizing measures

Michel J. G. Weber (2012)

Colloquium Mathematicae

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We give two examples of periodic Gaussian processes, having entropy numbers of exactly the same order but radically different small deviations. Our construction is based on Knopp's classical result yielding existence of continuous nowhere differentiable functions, and more precisely on Loud's functions. We also obtain a general lower bound for small deviations using the majorizing measure method. We show by examples that our bound is sharp. We also apply it to Gaussian independent sequences...

A geometric approach to correlation inequalities in the plane

A. Figalli, F. Maggi, A. Pratelli (2014)

Annales de l'I.H.P. Probabilités et statistiques

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By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt’s Gaussian correlation inequality (the two-dimensional case of the long-standing Gaussian correlation conjecture) is derived as a corollary,...