A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature

TL;DR

This paper extends Poisson-type deviation inequalities to positively curved Markov jump processes, providing generalized tail estimates and an application to birth-death processes, enhancing understanding of their probabilistic behavior.

Contribution

It introduces a new deviation inequality for Markov jump processes with positive Wasserstein curvature, generalizing previous tail estimates and applying to birth-death processes.

Findings

Derived a new Poisson-type deviation inequality.

Generalized tail estimates for Markov jump processes.

Applied results to birth-death processes.

Abstract

The purpose of this paper is to extend the investigation of Poisson-type deviation inequalities started by Joulin (Bernoulli 13 (2007) 782--798) to the empirical mean of positively curved Markov jump processes. In particular, our main result generalizes the tail estimates given by Lezaud (Ann. Appl. Probab. 8 (1998) 849--867, ESAIM Probab. Statist. 5 (2001) 183--201). An application to birth--death processes completes this work.