ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, 2008, Vol. 12, p. 12-29
DOI: 10.1051/ps:2007032

deviation bounds for additive functionals of markov processes

Patrick Cattiaux¹ and Arnaud Guillin²

¹ École Polytechnique, CMAP, 91128 Palaiseau cedex, France, CNRS 756, and Université Paris X Nanterre, équipe MODAL'X, UFR SEGMI, 200 avenue de la République, 92001 Nanterre cedex, France; cattiaux@cmapx.polytechnique.fr
² Ceremade, Université Paris IX Dauphine, 75775 Paris cedex, France, CNRS 7534; guillin@ceremade.dauphine.fr

Received March 13, 2006. Revised June 19 and September 26, 2006. Published online 13 November 2007

Abstract
In this paper we derive non asymptotic deviation bounds for

$\begin{displaymath}{\mathbb P}_\nu (\vert\frac 1t \int_0^t V(X_s) {\rm d}s - \int V {\rm d} \mu \vert \geq R)\end{displaymath}$

where X is a $\mu$ stationary and ergodic Markov process and V is some $\mu$ integrable function. These bounds are obtained under various moments assumptions for V, and various regularity assumptions for $\mu$ . Regularity means here that $\mu$ may satisfy various functional inequalities (F-Sobolev, generalized Poincaré etc.).

Mathematics Subject Classification. 60F10, 60J25

Key words: Deviation inequalities, functional inequalities, additive functionals.

© EDP Sciences, SMAI 2008