Polynomial deviation bounds for recurrent Harris processes having general state space

¹ CNRS UMR 8088, Département de Mathématiques, Université de Cergy-Pontoise, 95000 Cergy-Pontoise, France
eva.loecherbach@u-cergy.fr
² Département de Mathématiques, Université d’Evry-Val d’Essonne, Bd François Mitterrand, 91025 Evry Cedex, France
dasha.loukianova@univ-evry.fr

Received: 29 November 2010
Revised: 26 August 2011

Abstract

Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc et al. [Stoc. Proc. Appl. 119, (2009) 897–923] introduced verifiable conditions in terms of a supermartingale property implying an explicit control of modulated moments of hitting times. We show how this control can be translated into a control of polynomial moments of abstract regeneration times which are obtained by using the regeneration method of Nummelin, extended to the time-continuous context. As a consequence, if a p-th moment of the regeneration times exists, we obtain non asymptotic deviation bounds of the form Here, f is a bounded function and μ is the invariant measure of the process. We give several examples, including elliptic stochastic differential equations and stochastic differential equations driven by a jump noise.