A necessary and sufficient condition for the asymptotic normality of linear combinations of order statistics from the uniform distribution over [0, 1] is derived. The condition implies, that the variances of all weighted extremes are asymptotically zero compared with the total variance of the sum, which conversely, in the case of nonnegative constants, is sufficient for the asymptotic normality.
"A Characterization of the Asymptotic Normality of Linear Combinations of Order Statistics from the Uniform Distribution." Ann. Statist. 4 (6) 1244 - 1246, November, 1976. https://doi.org/10.1214/aos/1176343656