On probability of deviation of the sample mean from its median

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 $P_n$.