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Extremal Lipschitz functions in the deviation inequalities from the mean


 
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1. Title Title of document Extremal Lipschitz functions in the deviation inequalities from the mean
 
2. Creator Author's name, affiliation, country Dainius Dzindzalieta; Vilnius University; Lithuania
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Gaussian, vertex isoperimetric, deviation from the mean, inequalities, Hamming, probability metric space
 
3. Subject Subject classification Primary 60E15; Secondary 60A10.
 
4. Description Abstract We obtain an optimal deviation from the mean upper bound $D(x)=\sup\{\mu\{f-\mathbb{E}_{\mu} f\geq x\}:f\in\mathcal{F},x\in\mathbb{R}\}$ where $\mathcal{F}$ is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact bounds for Euclidean unit sphere $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\mathbb{R}^n$ equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance. We also prove that in general probability metric spaces the $\sup$ is achieved on a family of distance functions.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) This research was funded by a grant (No. MIP-12090) from the Research Council of Lithuania
 
7. Date (YYYY-MM-DD) 2013-08-06
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2814
 
10. Identifier Digital Object Identifier 10.1214/ECP.v18-2814
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 18
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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