TITLE: Tilted sum of independent random variables
QUESTION [5 upvotes]: Let $(X_i)_i$ be a sequence of centered i.i.d. random variables with finite variance. Is it true that
$$\frac{\sum_{i=1}^{\lfloor n^{0.6} \rfloor}X_i}{\sqrt{n}}\stackrel{\mbox{a.s.}}{\longrightarrow} 0\quad ?$$
Should be a sort of corollary of the central limit theorem but I don't know how to prove the a.s. convergence. Any help?

REPLY [3 votes]: Let $\alpha\in (0,1)$ and $m=\lfloor n^{\alpha} \rfloor$. Then
$$
\frac{1}{\sqrt{n}}\left|\sum_{i=1}^{\lfloor n^{\alpha} \rfloor}X_i\right|\le\frac{1}{m^{1/(2\alpha)}}\left|\sum_{i=1}^{m}X_i\right|\to 0 \quad\text{a.s.}
$$
by the Marcinkiewicz–Zygmund SLLN.