Measure concentration through non-Lipschitz observables and functional inequalities

TL;DR

This paper develops a general method to obtain non-Gaussian concentration estimates for invariant measures of reversible Markov processes, bypassing the need for Lipschitz observables by using functional inequalities and Lyapunov conditions.

Contribution

It introduces a novel approach combining functional inequalities with Lyapunov conditions to achieve concentration results without the Lipschitz assumption.

Findings

Applicable to diffusions and pure-jump Markov processes

Provides non-Gaussian concentration estimates

Circumvents classical Lipschitz requirement

Abstract

Non-Gaussian concentration estimates are obtained for invariant probability measures of reversible Markov processes. We show that the functional inequalities approach combined with a suitable Lyapunov condition allows us to circumvent the classical Lipschitz assumption of the observables. Our method is general and covers diffusions as well as pure-jump Markov processes on unbounded spaces.