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.