Polynomial deviation bounds for recurrent Harris processes having general state space
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 ν 1 t 0 t f(X s )ds-μ(f)K(p)1 t p-1 1 2(p-1) f 2(p-1) ,p2.
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.

DOI : 10.1051/ps/2011156
Classification : 60J55, 60J35, 60F10, 62M05
Mots clés : Harris recurrence, polynomial ergodicity, Nummelin splitting, continuous time Markov processes, drift condition, modulated moment
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     title = {Polynomial deviation bounds for recurrent {Harris} processes having general state space},
     journal = {ESAIM: Probability and Statistics},
     pages = {195--218},
     publisher = {EDP-Sciences},
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     year = {2013},
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     url = {http://archive.numdam.org/articles/10.1051/ps/2011156/}
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Löcherbach, Eva; Loukianova, Dasha. Polynomial deviation bounds for recurrent Harris processes having general state space. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 195-218. doi : 10.1051/ps/2011156. http://archive.numdam.org/articles/10.1051/ps/2011156/

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