Let $(X_i)_{i\geq 1}$ be iid zero mean and finite variances. Let $S_n=X_1+..+X_n$. What can we say about $\liminf\limits_{n\rightarrow \infty}\frac{|S_n|}{\sqrt{n}}?$
I can compute the result of $\liminf\limits_{n\rightarrow \infty}\frac{S_n}{\sqrt{n}}$ as follows:
Given $a$ be any real number, we have $$ P(\liminf \frac{S_n}{\sqrt{n}}>a)=\lim\limits_{n\rightarrow \infty} P(inf_{k\geq n} \frac{S_k}{\sqrt{k}}>a)\leq \lim\limits_{n\rightarrow \infty}P(\frac{S_n}{\sqrt{n}}>a)=P(Z>a)<1. $$ where $Z$ is $N(0,\sigma^2)$ by central limit theorem.
On the other hand, $\{\liminf \frac{S_n}{\sqrt{n}}>a\}$ is a tail event, and by 0-1 law, we conclude $$ P(\liminf \frac{S_n}{\sqrt{n}}>a)=0\ a.s,\ \mbox{which implies that}\ \ P(\liminf \frac{S_n}{\sqrt{n}}\leq a)=1\ \ a.s. $$ If we let $a$ decreases to $-\infty$, we obtain $$ \liminf \frac{S_n}{\sqrt{n}}=-\infty\ a.s. $$ But, I have no idea how to work on $\liminf \frac{|S_n|}{\sqrt{n}}$, and I am not sure whether the result I get is useful to solve it. Please give me some suggestion. Thanks.