Deviation Inequalities for Centered Additive Functionals of Recurrent Harris Processes Having General State Space

Abstract

Let X be a Harris recurrent strong Markov process in continuous time with general Polish state space E, having invariant measure μ. In this paper we use the regeneration method to derive non asymptotic deviation bounds for $$P_{x}\biggl(\biggl|\int_0^tf(X_s)\,ds\biggr|\geq t^{\frac{1}{2}+\eta}\varepsilon \biggr)$$ in the positive recurrent case, for nice functions f with μ(f)=0 (f must be a charge). We generalize these bounds to the fully null-recurrent case in the moderate deviations regime. We obtain a Gaussian concentration bound for all functions f which are a charge. The rate of convergence is expressed in terms of the deterministic equivalent of the process. The main ingredient of the proof is Nummelin splitting in continuous time, which allows one to introduce regeneration times for the process on an enlarged state space.