Let $E_n$ be an probability event which depends on $n$. Based on the book I read, Tao's Random Matrix Theory, an event $E_n$ holds asymptotically almost surely if it holds with probability $1-o(1)$, i.e., $$\textbf{P}(E_n) \ge 1 - o(1)$$
Suppose $X_i$'s are iid copies of $X$ where $\mathbf{E}X=0$ and $\mathbf{E}X^2=1$. Let $S_n = \sum_{i=1}^n X_i$. Then the weak law of large number says that for any $\epsilon > 0$, we have $$ \lim_{n\to \infty} \textbf{P}(|S_n| \le \epsilon n ) = 1 $$
From here, can we say that $$ S_n = o(1)n $$ asymptotically almost surely? I can see that for any fixed $\epsilon > 0$ (independent with $n$) $$ S_n = \epsilon n $$ holds asymptotically almost surely as the weak law of large number says $$ \textbf{P}(|S_n| \le \epsilon n ) = 1 - o(1) $$ But not sure whether it is possible to relate $\epsilon$ with $n$.
Any comments/suggestions/answers will be very appreciated. Thanks in advance.