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

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