Cramér moderate deviation expansion for martingales with one-sided Sakhanenko's condition and its applications

Abstract

We give a Cramér moderate deviation expansion for martingales with differences having finite conditional moments of order $2+\rho ,\rho \in (0,1],$ 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, it leads to a "half-side" moderate deviation principle for martingales. It is worth mentioning that our result is new even for independent random variables. Moreover, applications to quantile coupling inequality, $\beta$-mixing and $\psi$-mixing sequences are discussed.

24 pages

References in corpus (1)

• Cramér large deviation expansions for martingales under Bernstein's condition