Polynomial deviation bounds for recurrent Harris processes having general state space

Abstract

Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc, Fort and Guillin (2009) 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 $$P_{\nu}(|\frac1t\int_0^tf(X_s)ds-\mu(f)|\geq\ge)\leq K(p)\frac1{t^{p- 1}}\frac 1{\ge^{2(p-1)}}\|f\|_\infty^{2(p-1)}, p \geq 2. $$ Here, $f$ is a bounded function and $\mu$ 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.