Cram\'{e}r moderate deviation expansion for martingales with one-sided Sakhanenko's condition and its applications

TL;DR

This paper establishes a new Cramér moderate deviation expansion for martingales with specific moment conditions, leading to a half-side deviation principle and applications to mixing sequences and inequalities.

Contribution

It introduces a novel moderate deviation expansion for martingales under Sakhanenko's condition, applicable even to independent variables, with optimal bounds and broad applications.

Findings

Derived a new moderate deviation expansion for martingales.

Established a half-side moderate deviation principle.

Applied results to mixing sequences and quantile inequalities.

Abstract

We give a Cram\'{e}r 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.