Deviation inequalities for martingales with applications

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• 作者：Xiequan Fan^a ; ^b ; ^{fanxiequan@hotmail.com} ; Ion Grama^c ; ^{ion.grama@univ-ubs.fr} ; Quansheng Liu^c ; ^{quansheng.liu@univ-ubs.fr}
• 关键词：Large deviations ; Martingales ; Deviation inequalities ; Linear regressions ; Weak invariance principles ; Self-normalized deviations
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 April 2017
• 年：2017
• 卷：448
• 期：1
• 页码：538-566
• 全文大小：538 K
• 卷排序：448

文摘

Using changes of probability measure developed by Grama and Haeusler (2000) [18], we extend the deviation inequalities of Lanzinger and Stadtmüller (2000) [26] and Fuk and Nagaev (1971) [15] to the case of martingales. Our inequalities recover the best possible decaying rate in the independent case. In particular, these inequalities improve the results of Lesigne and Volný (2001) [27] under a stronger condition that the martingale differences have bounded conditional moments. Applications to linear regressions with martingale difference innovations, weak invariance principles for martingales and self-normalized deviations are provided. In particular, we establish a type of self-normalized deviation bounds for parameter estimation of linear regressions. Such type bounds have the advantage that they do not depend on the distribution of the regression random variables.