Metric and Mixing Sufficient Conditions for Concentration of Measure

Abstract

We derive sufficient conditions for a family $(X^n,\rho_n,P_n)$ of metric probability spaces to have the measure concentration property. Specifically, if the sequence $\{P_n\}$ of probability measures satisfies a strong mixing condition (which we call $\eta$-mixing) and the sequence of metrics $\{\rho_n\}$ is what we call $\Psi$-dominated, we show that $(X^n,\rho_n,P_n)$ is a normal Levy family. We establish these properties for some metric probability spaces, including the possibly novel $X=[0,1]$, $\rho_n=\ell_1$ case.