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 $$P_{\nu }(|\frac{1}{t}{\int }_0^{t}f(X_s)ds-\mu (f)|\ge \ge )\le K(p)\frac{1}{t^{p-1}}\frac{1}{{\ge }^{2(p-1)}}\Vert f{\Vert }_{\infty }^{2(p-1)},p\ge 2.$$ 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.

References in corpus (2)

• Subgeometric ergodicity of strong Markov processes
• Concentration for multidimensional diffusions and their boundary local times