Measure concentration for Euclidean distance in the case of dependent random variables
Abstract
Let q^n be a continuous density function in n-dimensional Euclidean space. We think of q^n as the density function of some random sequence X^n with values in \BbbR^n. For I\subset[1,n], let X_I denote the collection of coordinates X_i, i\in I, and let \bar X_I denote the collection of coordinates X_i, i\notin I. We denote by Q_I(x_I|\bar x_I) the joint conditional density function of X_I, given \bar X_I. We prove measure concentration for q^n 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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- October 2004
- DOI:
- 10.48550/arXiv.math/0410168
- arXiv:
- arXiv:math/0410168
- Bibcode:
- 2004math.....10168M
- Keywords:
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- Mathematics - Probability;
- 60K35;
- 82C22 (Primary)
- E-Print:
- Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000702