Geodesic
On probability of deviation of the sample mean from its median
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VIII, Tome 130 (1983), pp. 130-136

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Asymptotic behaviour of $P_n=\mathbf P(\bar X_n-\operatorname{med}\bar X_n\geqslant\varepsilon)$ is studied for i. i. d. r. v. $X_k$. Necessary and sufficient conditions are found for polinomial and exponential rates of convergence of $P_n$. 
@article{ZNSL_1983_130_a12,
     author = {A. I. Martikainen},
     title = {On probability of deviation of the sample mean from its median},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {130--136},
     publisher = {mathdoc},
     volume = {130},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_130_a12/}
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A. I. Martikainen. On probability of deviation of the sample mean from its median. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VIII, Tome 130 (1983), pp. 130-136. http://geodesic.mathdoc.fr/item/ZNSL_1983_130_a12/