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

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

Katalin Marton

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

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 X_I denote the collection of coordinates X_i, i∈I, and let $\widebar X_{I}$ denote the collection of coordinates X_i, i∉I. We denote by $Q_{I}(x_{I}|\bar{x}_{I})$ the joint conditional density function of X_I, 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 x_I, 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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Primary Subjects: 60K35, 82C22

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

Full-text: Open access

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Permanent link to this document: http://projecteuclid.org/euclid.aop/1091813622
Digital Object Identifier: doi:10.1214/009117904000000702
Mathematical Reviews number (MathSciNet): MR2078549
Zentralblatt MATH identifier: 02121705

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