Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Eva Löcherbach, Dasha Loukianova, and Oleg Loukianov

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Abstract

Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p>1, ∃a∈ℝ such that ∀x∈ℝ, $\mathbb{E}_{x}T_{a}^{p}\symbol{60}\infty$ and $\mathbb{E}_{\nu}T_{a}^{p/2}\symbol{60}\infty$, where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form

ν(|(1/t)0tf(Xs) dsμ(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 $\mathbb{E}_{x}T_{a}^{p}$ from above and below. This control is based on a generalized Kac’s formula (see Theorem 4.1) for the moments $\mathbb{E}_{x}f(T_{a})$ of a differentiable function f.

Résumé

Considérons une diffusion récurrente positive avec loi initiale ν et probabilité invariante μ. Pour tout a∈ℝ, soit Ta le temps d’atteinte du point a. Supposons qu’il existe p>1 et un point a∈ℝ tels que pour tout x∈ℝ, $\mathbb{E}_{x}T_{a}^{p}\symbol{60}\infty$ et $\mathbb{E}_{\nu}T_{a}^{p/2}\symbol{60}\infty$. Alors nous obtenons l’inégalité de déviation non-asymptotique suivante:

ν(|(1/t)0tf(Xs) dsμ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p,

f est une fonction bornée ou une fonction bornée à support compact. Ici, A(f)=‖f dans le cas d’une fonction bornée et A(f)=μ(|f|) dans le cas d’une fonction bornée à support compact.

De plus, sous certaines conditions sur les coefficients de la diffusion, nous obtenons une minoration et majoration, polynomiale en x, de $\mathbb{E}_{x}T_{a}^{p}$. Ce résultat est basé sur une formule de Kac généralisée (voir théoréme 4.1) pour les moments $\mathbb{E}_{x}f(T_{a})$f est une fonction dérivable.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 2 (2011), 425-449.

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1300887276

Digital Object Identifier
doi:10.1214/10-AIHP359

Mathematical Reviews number (MathSciNet)
MR2814417

Zentralblatt MATH identifier
1220.60045

Subjects
Primary: 60F99: None of the above, but in this section 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65]

Keywords
Diffusion process Recurrence Additive functionals Ergodic theorem Polynomial convergence Hitting times Kac formula Deviations inequalities

Citation

Löcherbach, Eva; Loukianova, Dasha; Loukianov, Oleg. Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 425--449. doi:10.1214/10-AIHP359. https://projecteuclid.org/euclid.aihp/1300887276


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