The Order of the Normal Approximation for Linear Combinations of Order Statistics with Smooth Weight Functions
Roelof Helmers · The Annals of Probability · 1977
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.