Vitesse dans le theoreme limite central pour certains processus stationnaires fortement decorreles

Vitesse de convergence dans le théorème limite central pour des chaînes de Markov fortement ergodiques

Let $Q$ be a transition probability on a measurable space $E$ which admits an invariant probability measure, let $(X_n)_n$ be a Markov chain associated to $Q$, and let $ξ$ be a real-valued measurable function on $E$, and $S_n=\sum _{k=1}^nξ(X_k)$. Under functional hypotheses on the action of $Q$ and the Fourier kernels $Q(t)$, we investigate the rate of convergence in the central limit theorem for the sequence $(\frac{S_n}{\sqrt{n}})_n$. According to the hypotheses, we prove that the rate is, either $\mathrm{O}(n^{-τ/{2}})$ for all $τ<1$, or $\mathrm{O}(n^{-{1}/{2}})$. We apply the spectral Na…