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

Löcherbach, Eva ; Loukianova, Dasha ; Loukianov, Oleg

Ann. Inst. H. Poincaré Probab. Statist., Tome 47 (2011) no. 1, p. 425-449 / Harvested from Project Euclid

Résumé

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 T_a is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form ¶ ℙ_ν(|(1/t)∫₀^tf(X_s) ds−μ(f)|≥ε)≤K(p)(1/t^p/2)(1/ε^p)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.

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Publié le : 2011-05-15
Classification:  Diffusion process,  Recurrence,  Additive functionals,  Ergodic theorem,  Polynomial convergence,  Hitting times,  Kac formula,  Deviations inequalities,  60F99,  60J55,  60J60