The Annals of Probability

Measure concentration for Euclidean distance in the case of dependent random variables

Katalin Marton

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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 $\widebar X_{I}$ 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 $\widebar X_{I}$. 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.

Article information

Source
Ann. Probab., Volume 32, Number 3B (2004), 2526-2544.

Dates
First available in Project Euclid: 6 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1091813622

Digital Object Identifier
doi:10.1214/009117904000000702

Mathematical Reviews number (MathSciNet)
MR2078549

Zentralblatt MATH identifier
1071.60012

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35]

Keywords
Measure concentration Wasserstein distance relative entropy Dobrushin–Shlosman mixing condition Gibbs sampler

Citation

Marton, Katalin. Measure concentration for Euclidean distance in the case of dependent random variables. Ann. Probab. 32 (2004), no. 3B, 2526--2544. doi:10.1214/009117904000000702. https://projecteuclid.org/euclid.aop/1091813622


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