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

Author

Listed:
  • Xiequan Fan

    (Tianjin University)

  • Ion Grama

    (Univ. Bretagne-Sud)

  • Quansheng Liu

    (Univ. Bretagne-Sud)

Abstract

We give a Cramér moderate deviation expansion for martingales with differences having finite conditional moments of order $$2+\rho , \rho \in (0,1]$$2+ρ,ρ∈(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, $$\beta $$β-mixing sequences and $$\psi $$ψ-mixing sequences are discussed.

Suggested Citation

  • Xiequan Fan & Ion Grama & Quansheng Liu, 2020. "Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications," Journal of Theoretical Probability, Springer, vol. 33(2), pages 749-787, June.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:2:d:10.1007_s10959-019-00949-2
    DOI: 10.1007/s10959-019-00949-2
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    References listed on IDEAS

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    1. Grama, Ion & Haeusler, Erich, 2000. "Large deviations for martingales via Cramér's method," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 279-293, February.
    2. Gao, Fu-Qing, 1996. "Moderate deviations for martingales and mixing random processes," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 263-275, February.
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