Almost Sure Invariance Principles via Martingale Approximation

Florence Merlevède; Costel Peligrad; Magda Peligrad

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arXiv · 1103.6266

Almost Sure Invariance Principles via Martingale Approximation

Florence Merlevède·Costel Peligrad·Magda Peligrad

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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, 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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Florence Merlevède, Costel Peligrad, Magda Peligrad. 2011-05-03. Almost Sure Invariance Principles via Martingale Approximation. https://arxiv.org/abs/1103.6266

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