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July 2004 Measure concentration for Euclidean distance in the case of dependent random variables
Katalin Marton
Ann. Probab. 32(3B): 2526-2544 (July 2004). DOI: 10.1214/009117904000000702

Abstract

Let qn be a continuous density function in n-dimensional Euclidean space. We think of qn as the density function of some random sequence Xn with values in $\Bbb{R}^{n}$. For I⊂[1,n], let XI denote the collection of coordinates Xi, iI, and let $\overline X_{I}$ denote the collection of coordinates Xi, iI. We denote by $Q_{I}(x_{I}|\bar{x}_{I})$ the joint conditional density function of XI, given $\overline X_{I}$. We prove measure concentration for qn in the case when, for an appropriate class of sets I, (i) the conditional densities $Q_{I}(x_{I}|\bar{x}_{I})$, as functions of xI, uniformly satisfy a logarithmic Sobolev inequality and (ii) these conditional densities also satisfy a contractivity condition related to Dobrushin and Shlosman’s strong mixing condition.

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Katalin Marton. "Measure concentration for Euclidean distance in the case of dependent random variables." Ann. Probab. 32 (3B) 2526 - 2544, July 2004. https://doi.org/10.1214/009117904000000702

Information

Published: July 2004
First available in Project Euclid: 6 August 2004

zbMATH: 1071.60012
MathSciNet: MR2078549
Digital Object Identifier: 10.1214/009117904000000702

Subjects:
Primary: 60K35 , 82C22

Keywords: Dobrushin–Shlosman mixing condition , Gibbs sampler , measure concentration , Relative entropy , Wasserstein distance

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3B • July 2004
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