In this paper, two versions of the Berry-Esseen theorems are established for $L$-statistics in the non-identically distributed case. One theorem, which requires $E|X_i|^3 < \infty$, is an extension of the classical Berry-Esseen theorem. Another, proved under the condition $E|X_i|^\alpha < \infty$ for some $\alpha \in (0, 1\rbrack$, seems to be of more interest for statistical inference. Some applications are also discussed.
"On the Berry-Esseen Bound for $L$-Statistics in the Non-I.D. Case with Applications to the Estimation of Location Parameters." Ann. Statist. 22 (2) 968 - 979, June, 1994. https://doi.org/10.1214/aos/1176325506