## The Annals of Probability

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

Katalin Marton

#### 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 . For I⊂[1,n], let XI denote the collection of coordinates Xi, iI, and let denote the collection of coordinates Xi, iI. We denote by the joint conditional density function of XI, given . We prove measure concentration for qn in the case when, for an appropriate class of sets I, (i) the conditional densities , 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

https://projecteuclid.org/euclid.aop/1091813622

Digital Object Identifier
doi:10.1214/009117904000000702

Mathematical Reviews number (MathSciNet)
MR2078549

Zentralblatt MATH identifier
1071.60012

#### 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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