Almost sure invariance principles via martingale approximation |
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Authors: | Florence Merlevède |
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Institution: | a Université Paris Est, Laboratoire de mathématiques, UMR 8050 CNRS, Bâtiment Copernic, 5 Boulevard Descartes, 77435 Champs-Sur-Marne, Franceb Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA |
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Abstract: | In 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. |
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Keywords: | primary 60F05 60F15 60J05 |
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