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.

Keywords: concentration of measure, martingale differences, metric probability space, Levy family, strong mixing

References in corpus (2)

• Measure Concentration of Hidden Markov Processes
• Measure Concentration of Markov Tree Processes

Cited by in corpus (3)

• Measure Concentration of Hidden Markov Processes
• Measure Concentration of Markov Tree Processes
• A Linear Programming Inequality with Applications to Concentration of Measure