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

The purpose of this paper is to extend the investigation of Poisson-type deviation inequalities started by Joulin (Benioulli 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 (Anti. Appl. Probab. 8 (1998) 849-867, ESAIM Probab. Statist. 5 (2001) 183-201). An application to birth-death processes completes this work.