Abstract
A Berry-Esseen bound of order $n^{-\frac{1}{2}}$ is established for linear combinations of order statistics. The theorem requires a "smooth" weight function, and the underlying distribution function must not have "too much weight in the tails." The distribution function need not be continuous.
Citation
R. Helmers. "The Order of the Normal Approximation for Linear Combinations of Order Statistics with Smooth Weight Functions." Ann. Probab. 5 (6) 940 - 953, December, 1977. https://doi.org/10.1214/aop/1176995662
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