Concentration Inequalities for Markov Jump Processes

TL;DR

This paper develops new concentration inequalities for empirical means of irreducible Markov jump processes, utilizing multiple mathematical techniques to provide bounds and a Bernstein-type inequality.

Contribution

It introduces a general framework for concentration inequalities for Markov jump processes using Feynman-Kac semigroups and three distinct methods.

Findings

Derived a general concentration inequality for empirical means.

Established a Bernstein-type concentration inequality.

Applied perturbation, Poincaré, and information inequalities.

Abstract

We derive concentration inequalities for empirical means $\frac{1}{t} \int_0^t f(X_s) ds$ where $X_s$ is an irreducible Markov jump process on a finite state space and $f$ some observable. Using a Feynman-Kac semigroup we first derive a general concentration inequality. Then, based on this inequality we derive further concentration inequalities. Hereby we use three different methods; perturbation theory, Poincar\'e inequalities and information inequalities. We also obtain a Bernstein type concentration inequality.