One-sided versions of strong limit theorems

Let $\{S_n\}$ be a random walk, $T_n=S_n-\operatorname{med}(S_n)$, $1/2<\beta<\infty$. Necessary and sufficient conditions for
\begin{gather*} \limsup T_n/n^\beta=0\quad \text{a.\,s.} \\ \limsup T_n/n^\beta=1\quad \text{a.\,s.} \\ \limsup T_n/(2n\log\log n)^{1/2}=1\quad \text{a.\,s.} \end{gather*}
are obtained.