Polynomial deviation bounds for recurrent Harris processes having general state space

Löcherbach, Eva; Loukianova, Dasha

ESAIM: Probability and Statistics, Tome 17 (2013), pp. 195-218

Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc et al. [Stoc. Proc. Appl. 119, (2009) 897-923] 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

$$P_{\nu }\left(\left|\frac{1}{t}{\int }_0^{t}f\left(X_s\right)\mathrm{d}s-\mu \left(f\right)\right|\ge \ge \right)\le K\left(p\right)\frac{1}{t^{p-1}}\frac{1}{{\ge }^{2(p-1)}}{\parallel f\parallel }_{\infty }^{2(p-1)},p\ge 2.$$

P ν 1 t ∫ 0 t f ( X s ) d s - μ ( f ) ≥ ε ≤ K ( p ) 1 t p - 1 1 ε 2 ( p - 1 ) ∥ f ∥ ∞ 2 ( p - 1 ) , p ≥ 2. Here, f is a bounded function and μ 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.

Classification : 60J55, 60J35, 60F10, 62M05
Keywords: Harris recurrence, polynomial ergodicity, Nummelin splitting, continuous time Markov processes, drift condition, modulated moment

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