Almost sure invariance principles via martingale approximation

Florence Merlevede (1) , Costel Peligrad , Magda Peligrad

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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. (C) 2011 Elsevier B.V. All rights reserved.

Domaines: Physique mathématique [math-ph]

Florence Merlevede, Costel Peligrad, Magda Peligrad. Almost sure invariance principles via martingale approximation. Stochastic Processes and their Applications, 2012, 122 (1), pp.170--190. ⟨10.1016/j.spa.2011.09.004⟩. ⟨hal-00692710⟩