# Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times

• Volume: 47, Issue: 2, page 425-449
• ISSN: 0246-0203

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## Abstract

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Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p&gt;1, ∃a∈ℝ such that ∀x∈ℝ, and , where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙν(|(1/t)∫0tf(Xs) ds−μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p. Here f bounded or bounded and compactly supported and A(f)=‖f‖∞ when f is bounded and A(f)=μ(|f|) when f is bounded and compactly supported. We also give, under some conditions on the coefficients of X, a polynomial control of from above and below. This control is based on a generalized Kac’s formula (see Theorem 4.1) for the moments of a differentiable function f.

## How to cite

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Löcherbach, Eva, Loukianova, Dasha, and Loukianov, Oleg. "Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times." Annales de l'I.H.P. Probabilités et statistiques 47.2 (2011): 425-449. <http://eudml.org/doc/242012>.

@article{Löcherbach2011,
abstract = {Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p&gt;1, ∃a∈ℝ such that ∀x∈ℝ, and , where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙν(|(1/t)∫0tf(Xs) ds−μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p. Here f bounded or bounded and compactly supported and A(f)=‖f‖∞ when f is bounded and A(f)=μ(|f|) when f is bounded and compactly supported. We also give, under some conditions on the coefficients of X, a polynomial control of from above and below. This control is based on a generalized Kac’s formula (see Theorem 4.1) for the moments of a differentiable function f.},
author = {Löcherbach, Eva, Loukianova, Dasha, Loukianov, Oleg},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {diffusion process; recurrence; additive functionals; ergodic theorem; polynomial convergence; hitting times; Kac formula; deviations inequalities},
language = {eng},
number = {2},
pages = {425-449},
publisher = {Gauthier-Villars},
title = {Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times},
url = {http://eudml.org/doc/242012},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Löcherbach, Eva
AU - Loukianova, Dasha
AU - Loukianov, Oleg
TI - Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 425
EP - 449
AB - Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p&gt;1, ∃a∈ℝ such that ∀x∈ℝ, and , where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙν(|(1/t)∫0tf(Xs) ds−μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p. Here f bounded or bounded and compactly supported and A(f)=‖f‖∞ when f is bounded and A(f)=μ(|f|) when f is bounded and compactly supported. We also give, under some conditions on the coefficients of X, a polynomial control of from above and below. This control is based on a generalized Kac’s formula (see Theorem 4.1) for the moments of a differentiable function f.
LA - eng
KW - diffusion process; recurrence; additive functionals; ergodic theorem; polynomial convergence; hitting times; Kac formula; deviations inequalities
UR - http://eudml.org/doc/242012
ER -

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