Let $X_1, X_2, ...$ be independent, identically distributed $L^2$ random variables (i.e. finite variance); let $m_n$ denote any median of $S_n = \sum_{j = 1} ^ n X_j$ for $n \ge 1$. I am to show that for $\alpha > \frac 1 2$, we have a strong law:
$\displaystyle \frac{S_n - m_n}{n^\alpha} \stackrel{a.s.}{\to} 0$
where $\stackrel{a.s.}{\to}$ denotes almost sure convergence.
My thought process: Well, I'm not sure where $L^2$ comes into the picture. I've been thinking of trying to show something of the form $m_n - nE(X_1) = o(n^\alpha)$, since then I can apply the strong law (this being where the condition that $\alpha > 1/2$ would come in since I need that for the strong law to hold). I'm guessing $L^2$'ness comes into play by helping to get $m_n$ and $nE(X_1)$ close enough together? But I'm not sure how. This also might be completely off point.