Optimal concentration inequalities for dynamical systems

TL;DR

This paper establishes optimal exponential and polynomial concentration inequalities for dynamical systems modeled by Young towers, depending on tail behavior, with applications to specific systems and observables.

Contribution

It provides the first sharp concentration inequalities for dynamical systems with different tail behaviors, extending previous results to a broader class of systems.

Findings

Exponential concentration inequalities for systems with exponential tails.

Polynomial concentration inequalities for systems with polynomial tails.

Applications demonstrating the inequalities' effectiveness in specific dynamical systems.

Abstract

For dynamical systems modeled by a Young tower with exponential tails, we prove an exponential concentration inequality for all separately Lipschitz observables of n variables. When tails are polynomial, we prove polynomial concentration inequalities. Those inequalities are optimal. We give some applications of such inequalities to specific systems and specific observables.