Translator Disclaimer
2013 Extremal Lipschitz functions in the deviation inequalities from the mean
Dainius Dzindzalieta
Author Affiliations +
Electron. Commun. Probab. 18: 1-5 (2013). DOI: 10.1214/ECP.v18-2814

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.

Citation

Download Citation

Dainius Dzindzalieta. "Extremal Lipschitz functions in the deviation inequalities from the mean." Electron. Commun. Probab. 18 1 - 5, 2013. https://doi.org/10.1214/ECP.v18-2814

Information

Accepted: 6 August 2013; Published: 2013
First available in Project Euclid: 7 June 2016

zbMATH: 1376.60041
MathSciNet: MR3091724
Digital Object Identifier: 10.1214/ECP.v18-2814

Subjects:
Primary: 60E15
Secondary: 60A10

JOURNAL ARTICLE
5 PAGES


SHARE
Back to Top