Théorème local pour chaînes de Markov de probabilité de transition quasi-compacte. Applications aux chaînes V-géométriquement ergodiques et aux modèles itératifs

Let Q be a transition probability on a measurable space E, let (Xn)n be a Markov chain associated to Q, and let ξ be a real-valued measurable function on E such that (ξ(Xn))n⩾0 satisfies a central limit theorem. Under functional hypotheses on the action of Q and its Fourier kernels Q(t), we establish a local limit theorem. We use the spectral method of Nagaev improved by a perturbation theorem of Keller and Liverani. The conditions required here on Q(t) are weaker than the ones usually imposed when the standard perturbation theorem is used in the spectral method. We give some applications to V-geometric ergodic chains and to Lipschitz iterative models.