Polynomial deviation bounds for recurrent Harris processes having general state space

Eva Löcherbach; Dasha Loukianova

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 195-218
  • ISSN: 1292-8100

Abstract

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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 ) d s - μ ( f ) ≥ ≥ ≤ K ( p ) 1 t p - 1 1 ≥ 2 ( p - 1 ) ∥ f ∥ ∞ 2 ( p - 1 ) , p ≥ 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.

How to cite

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Löcherbach, Eva, and Loukianova, Dasha. "Polynomial deviation bounds for recurrent Harris processes having general state space." ESAIM: Probability and Statistics 17 (2013): 195-218. <http://eudml.org/doc/273616>.

@article{Löcherbach2013,
abstract = {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\begin\{equation*\} P\_\{\nu \} \left(\left|\frac\{1\}\{t\}\int \_0^tf(X\_s)\{\rm d\}s-\mu (f)\right|\ge \ge \right)\le K(p)\frac\{1\}\{t^\{p- 1\}\}\frac\{1\}\{\ge ^\{2(p-1)\}\}\Vert f\Vert \_\infty ^\{2(p-1)\} ,\quad p \ge 2. \end\{equation*\}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.},
author = {Löcherbach, Eva, Loukianova, Dasha},
journal = {ESAIM: Probability and Statistics},
keywords = {Harris recurrence; polynomial ergodicity; Nummelin splitting; continuous time Markov processes; drift condition; modulated moment; polynomial deviation bounds},
language = {eng},
pages = {195-218},
publisher = {EDP-Sciences},
title = {Polynomial deviation bounds for recurrent Harris processes having general state space},
url = {http://eudml.org/doc/273616},
volume = {17},
year = {2013},
}

TY - JOUR
AU - Löcherbach, Eva
AU - Loukianova, Dasha
TI - Polynomial deviation bounds for recurrent Harris processes having general state space
JO - ESAIM: Probability and Statistics
PY - 2013
PB - EDP-Sciences
VL - 17
SP - 195
EP - 218
AB - 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\begin{equation*} P_{\nu } \left(\left|\frac{1}{t}\int _0^tf(X_s){\rm d}s-\mu (f)\right|\ge \ge \right)\le K(p)\frac{1}{t^{p- 1}}\frac{1}{\ge ^{2(p-1)}}\Vert f\Vert _\infty ^{2(p-1)} ,\quad p \ge 2. \end{equation*}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.
LA - eng
KW - Harris recurrence; polynomial ergodicity; Nummelin splitting; continuous time Markov processes; drift condition; modulated moment; polynomial deviation bounds
UR - http://eudml.org/doc/273616
ER -

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