Almost sure invariance principles via martingale approximation

作者：Merlevede Florence; Peligrad Costel; Peligrad Magda^*

来源：Stochastic Processes and Their Applications, 2012, 122(1): 170-190.

DOI：10.1016/j.spa.2011.09.004

• Martingale approximation
• Quenched CLT
• Almost sure CLT
• Normal Markov chains
• Functional CLT
• Law of the iterated logarithm
• Almost sure approximation

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.

• 出版日期2012-1