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.

中文翻译：

---

单面Sakhanenko条件鞅的Cramér中度偏差扩展及其应用

我们为鞅给出了一个 Cramér 中等偏差展开式，其差分具有有限条件矩 $$2+\rho , \rho \in (0,1]$$ 2 + ρ , ρ ∈ ( 0 , 1 ] , 和有限一-边条件指数矩。有效范围的上限和我们扩展的剩余部分都是最优的。因此，我们的结果导致了鞅的单边适度偏差原则。此外，应用于分位数耦合不等式，$$\讨论了 beta $$ β-混合序列和 $$\psi $$ ψ-混合序列。