Geodesic
The central limit theorem under the absence of extremal absolute order statistics
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 59-67

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One finds conditions for the relation $\Delta_{n,r}=o(1)=\nabla_{n,r}$, где $\Delta_{n,r}=\sup_x|P\Big(\frac{S_{n,r}}{a_n}$, $S_{n,r}=X_{(1)}+\dots+X_{(n-r)}$, $X_{(1)},\dots,X_{(n)}$ are the absolute order statistics for a repeated sample from a symmetric distribution. 
@article{ZNSL_1985_142_a5,
     author = {V. A. Egorov},
     title = {The central limit theorem under the absence of extremal absolute order statistics},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {59--67},
     publisher = {mathdoc},
     volume = {142},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a5/}
}
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V. A. Egorov. The central limit theorem under the absence of extremal absolute order statistics. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 59-67. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a5/