Almost sure invariance principles via martingale approximation

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• 作者：Florence  ; Merlevè ; de^a ;   ; ^{florence.merlevede@univ-mlv.fr} ; Costel  ; Peligrad^b ;   ; ^{peligrc@ucmail.uc.edu} ; Magda  ; Peligrad^b ;   ; ^{peligrm@ucmail.uc.edu}
• 关键词：Martingale approximation ; Quenched CLT ; Almost sure CLT ; Normal Markov chains ; Functional CLT ; Law of the iterated logarithm ; Almost sure approximation
• 刊名：Stochastic Processes and their Applications
• 出版年：2012
• 出版时间：January 2012
• 年：2012
• 卷：122
• 期：1
• 页码：170-190
• 全文大小：263 K

文摘

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.