Metric and Mixing Sufficient Conditions for Concentration of Measure

TL;DR

This paper establishes sufficient conditions involving mixing properties and metric domination that ensure measure concentration in families of metric probability spaces, including novel cases like the unit interval with b1-norm.

Contribution

It introduces the b5-mixing and a-dominated conditions as new criteria for measure concentration, extending the class of spaces known to exhibit this property.

Findings

Proves measure concentration under b5-mixing and a-dominated conditions.

Includes the b1-norm space [0,1] as a new example.

Shows these conditions imply the spaces form a normal Levy family.

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.