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

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 $\overline X_{I}$ 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 $\overline X_{I}$. 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.