Abstract
Cramér’s moderate deviations give a quantitative estimate for the relative error of the normal approximation and provide theoretical justifications for many estimators used in statistics. In this paper, we establish self-normalized Cramér type moderate deviations for martingales under some mild conditions. The result extends an earlier work of Fan et al. (Bernoulli 25 (2019) 2793–2823). Moreover, applications of our result to Student’s statistic, stationary martingale difference sequences and branching processes in a random environment are also discussed.
Funding Statement
Fan was partially supported by the National Natural Science Foundation of China (Grant Nos. 12371155 and 11971063). Shao was partially supported by the National Natural Science Foundation of China (Grant No. 12031005) and Shenzhen Outstanding Talents Training Fund.
Acknowledgements
The authors are deeply indebted to the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper. Fan would like to thank Quansheng Liu for his helpful discussion on the harmonic moments for branching processes in a random environment.
Citation
Xiequan Fan. Qi-Man Shao. "Self-normalized Cramér type moderate deviations for martingales and applications." Bernoulli 31 (1) 130 - 161, February 2025. https://doi.org/10.3150/24-BEJ1722
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