Deviation inequalities for centered additive functionals of recurrent Harris processes having general state space

TL;DR

This paper establishes deviation inequalities for additive functionals of Harris recurrent Markov processes with general state spaces, extending results to null-recurrent cases and providing Gaussian concentration bounds.

Contribution

It introduces non-asymptotic deviation bounds for additive functionals of Harris recurrent processes, including null-recurrent cases, using regeneration methods and Nummelin splitting.

Findings

Derived deviation bounds for positive recurrent Harris processes

Extended bounds to null-recurrent processes in moderate deviations regime

Established Gaussian concentration bounds for charge functions

Abstract

Let $X$ be a Harris recurrent strong Markov process in continuous time with general Polish state space $E,$ having invariant measure $\mu .$ In this paper we use the regeneration method to derive non asymptotic deviation bounds for $$P_{x} (|\int_0^tf(X_s)ds|\geq t^{\frac12 + \eta} \ge)$$ in the positive recurrent case, for nice functions $f$ with $\mu (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 contentration 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 to introduce regeneration times for the process on an enlarged state space.