  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.
