https://hal.archives-ouvertes.fr/hal-00692710Merlevede, FlorenceFlorenceMerlevedeLAMA - Laboratoire d'Analyse et de Mathématiques Appliquées - UPEM - Université Paris-Est Marne-la-Vallée - Fédération de Recherche Bézout - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12 - CNRS - Centre National de la Recherche ScientifiquePeligrad, CostelCostelPeligradPeligrad, MagdaMagdaPeligradAlmost sure invariance principles via martingale approximationHAL CCSD2012[MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]Lama, Admin2012-05-01 12:19:592022-09-29 14:21:152012-05-01 12:19:59enJournal articles10.1016/j.spa.2011.09.0041In this paper, we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, the almost sure central limit theorem, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; they are easy to verify, for instance, for linear processes and functions of Bernoulli shifts. (C) 2011 Elsevier B.V. All rights reserved.