Let $X_1,X_2,...$ be iid random variables with $p:=P(X=1)$ and $q:=P(X=-1)$ such that $p+q=1$. I want to study the behavior of the random walk $(S_n)_{n\geq 0}$ defined as $S_n=X_1+\cdots+X_n$, with $S_0=0$, when $n\to\infty$.
I know, by strong law of large numbers, that $S_n\overset{a.s.}{\to}\infty$ when $p>q$ and $S_n\overset{a.s.}{\to}-\infty$ when $p<q$.
What happens when $p=q=1/2$? In this case I can prove that $\lim_{n\to\infty}P(S_n\in[a,b])=0$ for all $a<b$.
Also In Random Walk Limit Behavior is mentioned that $\liminf S_n=-\infty$ and $\limsup S_n=\infty$ a.s.
Is that all that can be said? Is it true that $P(\lim_{n\to\infty}S_n\in[a,b])=0$ for all $a<b$?
The behavior at $n\rightarrow \infty$ is exactly the same as at $n=10$ or $n=1000$ $S_n$ will pass through $0$ a lot of times