https://hal-cnrs.archives-ouvertes.fr/hal-03241597v2Françoise, PènePèneFrançoiseLMBA - Laboratoire de Mathématiques de Bretagne Atlantique - UBS - Université de Bretagne Sud - UBO - Université de Brest - CNRS - Centre National de la Recherche ScientifiqueProbabilistic limit theorems via the operator perturbation method, under optimal momentassumptionsHAL CCSD2022Markov chainsMarkov random walkscentral limit theoremslocal limit theoremsEdgeworth expansionspectral method[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]Pene, Francoise2022-08-05 15:18:142022-08-06 03:36:552022-08-05 15:26:40enPreprints, Working Papers, ...https://hal-cnrs.archives-ouvertes.fr/hal-03241597v1application/pdf2The Nagaev-Guivarc’h operator perturbation method is well known to provide various probabilistic limit theorems for Markov random walks. A natural conjecture is that this method should provide these limit theorems under the same moment assumptions as the optimal ones in the case of sums of independent and identically distributed random variables. In the past decades, assumptions have been aweaken, without achieving fully this purpose (achievingit either with the help of an extra proof of the central limit theorem, or with an additional ε in the moment assumptions). The aim of this article is to give a positive answer to this conjecture via the Keller-Liverani theorem. We present here an approach allowing the establishment of limit theorems (including higher order ones) under optimal moment assumptions. Our method is based on Taylor expansions obtained via the perturbation operator method, combined witha new weak compacity argument without the use of any other extra tool (such as Martingale decomposition method, etc.).