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Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications

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Abstract

We give a Cramér moderate deviation expansion for martingales with differences having finite conditional moments of order \(2+\rho , \rho \in (0,1]\), and finite one-sided conditional exponential moments. The upper bound of the range of validity and the remainder of our expansion are both optimal. Consequently, our result leads to a one-sided moderate deviation principle for martingales. Moreover, applications to quantile coupling inequality, \(\beta \)-mixing sequences and \(\psi \)-mixing sequences are discussed.

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Acknowledgements

We would like to thank the associate editor for his helpful suggestions on the structure of paper. We also deeply indebted to the anonymous referees for their helpful comments. Fan and Liu have been partially supported by the National Natural Science Foundation of China (Grant nos. 11601375, 11626250, 11571052, 11401590 and 11731012), and by Hunan Natural Science Foundation (China, Grant no. 2017JJ2271). Grama and Liu have benefitted from the support of the French government “Investissements d’Avenir” program ANR-11-LABX-0020-01.

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Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Xiequan Fan

  2. UMR 6205, LMBA, Univ. Bretagne-Sud, 56000, Vannes, France

    Ion Grama & Quansheng Liu

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  1. Xiequan Fan
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  2. Ion Grama
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Correspondence to Xiequan Fan.

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Fan, X., Grama, I. & Liu, Q. Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications. J Theor Probab 33, 749–787 (2020). https://doi.org/10.1007/s10959-019-00949-2

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