Comments on the rate of convergence to asymptotic independence between order statistics

The main purpose of this paper is to correct the statement of Theorem 3.1 of Werner Hürlimann [2004. Statist. Probab. Lett. 66, 355-362]. This theorem yields a non-parametric criterion of asymptotically independence between order statistics. A very short proof of the corrected statement is given. Moreover, as a consequence of this theorem a simple non-parametric criterion is presented to unify known results about the asymptotic independence between order statistics.