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Large deviations for martingales via Cramér's method

Author

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  • Grama, Ion
  • Haeusler, Erich

Abstract

We develop a new approach for proving large deviation results for martingales based on a change of probability measure. It extends to the case of martingales the conjugate distribution technique due to Cramér. To demonstrate our approach, we derive formulae for probabilities of large deviations for martingales with bounded jumps and bounded norming factor. Surprisingly enough, our result shows that the relative error in the normal range is of the same order as in the case of sums of independent random variables. It also allows to extend the range beyond the normal one.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:85:y:2000:i:2:p:279-293
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    Citations

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    Cited by:

    1. 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.
    2. Fan, Xiequan & Grama, Ion & Liu, Quansheng & Shao, Qi-Man, 2020. "Self-normalized Cramér type moderate deviations for stationary sequences and applications," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5124-5148.
    3. Fan, Xiequan & Ma, Xiaohui, 2020. "On the Wasserstein distance for a martingale central limit theorem," Statistics & Probability Letters, Elsevier, vol. 167(C).
    4. Fan, Xiequan, 2017. "Self-normalized deviation inequalities with application to t-statistic," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 158-164.
    5. Sason, Igal, 2013. "Tightened exponential bounds for discrete-time conditionally symmetric martingales with bounded jumps," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1928-1936.
    6. Kanaya, Shin & Otsu, Taisuke, 2012. "Large deviations of realized volatility," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 546-581.
    7. Fan, Xiequan & Grama, Ion & Liu, Quansheng, 2012. "Hoeffding’s inequality for supermartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3545-3559.
    8. Dasgupta, Amites, 2024. "Azuma-Hoeffding bounds for a class of urn models," Statistics & Probability Letters, Elsevier, vol. 204(C).

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