Polynomial bounds in the Ergodic Theorem for positive recurrent one-dimensional diffusions and integrability of hitting times

Abstract

Let $X$ be a one dimensional positive recurrent diffusion with initial distribution $\nu$ and invariant probability $\mu$. Suppose that for some $p>1$, $\exists a\in \mathbb{R}$ such that $\forall x\in\R, \E_x T_a^p<\infty$ and $\E_νT_a^{p/2}<\infty$, where $T_a$ is the hitting time of $a$. For such a diffusion, we derive non asymptotic deviation bounds of the form $${\P }_{\nu }(|\frac{1}{t}{\int }_0^{t}f(X_s)ds-\mu (f)|\ge \ge )\le K(p)\frac{1}{t^{p/2}}\frac{1}{{\ge }^p}A(f{)}^p.$$ Here $f$ bounded or bounded and compactly supported and $A(f)=\Vert f{\Vert }_{\infty }$ when $f$ is bounded and $A(f)=\mu (|f|)$ when $f$ is bounded and compactly supported. We also give, under some conditions on the coefficients of $X$, a polynomial control of $\E_xT_a^p$ from above and below. This control is based on a generalized Kac's formula (see theorem \ref{thm:mainKac}) for the moments $\E_x f(T_a)$ of a differentiable function $f$.

References in corpus (3)

• Subgeometric ergodicity of strong Markov processes
• Poincaré inequality and exponential integrability of hitting times for linear diffusions
• On the Bennett-Hoeffding inequality

Cited by in corpus (1)

• Poincaré inequality and exponential integrability of hitting times for linear diffusions