Vitesse de convergence dans le th\'{e}or\`{e}me limite central pour des cha\^{i}nes de Markov fortement ergodiques

TL;DR

This paper investigates the rate of convergence in the central limit theorem for strongly ergodic Markov chains, improving spectral methods with perturbation theorems to handle weaker conditions and establish convergence rates.

Contribution

It introduces an enhanced spectral approach using Keller and Liverani's perturbation theorem to obtain optimal convergence rates under weaker moment conditions.

Findings

Convergence rate is either O(n^{-τ/2}) for all τ<1 or O(n^{-1/2}).

Spectral Nagaev's method combined with perturbation improves results.

Weaker moment conditions suffice for convergence in non-compact or unbounded cases.

Abstract

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 $\xi$ be a real-valued measurable function on $E$, and $S_n=\sum _{k=1}^n\xi(X_k)$. 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 $\mathrm{O}(n^{-{\tau}/{2}})$ for all $\tau<1$, or $\mathrm{O}(n^{-{1}/{2}})$. We apply the spectral Nagaev's method which is improved by using a perturbation theorem of Keller and Liverani, and a majoration of $|\mathbb{E}[\mathrm{e}^{\mat hrm{i}t{S_n}/{\sqrt{n}}}]-\mathrm{e}^{{-t^2}/{2}}|$ obtained by a method of martingale difference reduction. When $E$ is not compact or $\xi$ is not bounded, the conditions required here on $Q(t)$ (in substance, some moment conditions on $\xi$) are weaker than the ones usually imposed when the standard perturbation theorem is used in the spectral method. For example, in the case of $V$-geometric ergodic chains or Lipschitz iterative models, the rate of convergence in the c.l.t. is $\mathrm{O}(n^{-{1}/{2}})$ under a third moment condition on $\xi$.