Cram\'er'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 Cram\'{e}r 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 Cram\'{e}r type moderate deviations for Student's $t$-statistic for branching processes in a random environment.

翻译：Cramér型中偏差给出了正态逼近相对误差的定量估计，并为统计学中使用的许多估计量提供了理论依据。本文在较温和条件下建立了鞅的自正则Cramér型中偏差，扩展了Fan、Grama、Liu和Shao [Bernoulli， 2019] 的早期工作。此外，我们还讨论了该结果在Student统计量、平稳鞅差序列及随机环境分枝过程中的应用。特别地，我们建立了随机环境分枝过程中Student $t$统计量的Cramér型中偏差。