Polynomial deviation bounds for recurrent Harris processes having general state space

TL;DR

This paper establishes polynomial deviation bounds for recurrent Harris processes in general state spaces by linking supermartingale conditions to control of regeneration times, leading to explicit non-asymptotic deviation bounds.

Contribution

It extends the control of hitting times to polynomial moments of regeneration times using Nummelin's method in continuous time, providing new deviation bounds for Harris processes.

Findings

Derived explicit deviation bounds depending on the p-th moment of regeneration times

Applied results to elliptic SDEs and jump noise SDEs

Demonstrated the bounds with several process examples

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.