Strong moderate deviation theorems are concerned with relative errors in the tails caused by replacing the exact distribution function by its limiting distribution function. A new approach for deriving such theorems is presented using strong approximation inequalities. In this way a strong moderate deviation theorem is obtained for statistics of the form $T(\alpha_n)$, where $T$ is a sublinear functional and $\alpha_n$ is the empirical process. The basic theorem is also applied on linear combinations of order statistics, leading to a substantial improvement of previous results.
Tadeusz Inglot. Wilbert C. M. Kallenberg. Teresa Ledwina. "Strong Moderate Deviation Theorems." Ann. Probab. 20 (2) 987 - 1003, April, 1992. https://doi.org/10.1214/aop/1176989814