Self-normalized Cramér type moderate deviations for martingales and applications

09/11/2023

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by   Xiequan Fan, et al.

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Cramér's moderate deviations give a quantitative estimate for the relative error of the normal approximation and provide theoretical justifications for many estimator used in statistics. In this paper, we establish self-normalized Cramér type moderate deviations for martingales under some mile conditions. The result extends an earlier work of Fan, Grama, Liu and Shao [Bernoulli, 2019]. Moreover, applications of our result to Student's statistic, stationary martingale difference sequences and branching processes in a random environment are also discussed. In particular, we establish Cramér type moderate deviations for Student's t-statistic for branching processes in a random environment.