Using Central Limit Theorem to show that random walk exits a interval a.s. in finite time. Let $X_0 = x \in \mathbb{Z}$ and $X_1, X_2, \dots$ are i.i.d. random variables with values in $\{-1,0,1\}$ all with positive probability and $E(X_1) = 0$. Let $\sigma^2 = E(X_1^2)$.
Let $S_n = \sum_{0=1}^n X_n = x + \sum_{i=1}^n X_n$. Then $\frac{1}{\sqrt{n}} S_n \to N(x, \sigma^2)$ by the CLT.
How do we use this to show that for a interval $(a,b)$ with $a < x < b$, $S_n$ exits $(a,b)$ in finite time almost surely.
I thought of the following
$ \lim_{n \to \infty} P(S_n \in (a,b)) = \lim_{n \to \infty} P(\frac{S_n - x}{\sigma \sqrt{n}} \in (\frac{a - x}{\sigma \sqrt{n}}, \frac{b - x}{\sigma \sqrt{n}})) = \lim_{n \to \infty} \Phi(\frac{a - x}{\sigma \sqrt{n}}) - \Phi(\frac{b - x}{\sigma \sqrt{n}}) = \Phi(0) - \Phi(0) = 0 $
But I'm not convinced because the intervals depend on $n$.
Any ideas on how to make this precise, or this OK? Thanks.
 A: Question: What are the domains of the random variables $X_1, X_2, \ldots$? I would assume discrete time steps and thus the range is $\mathbb N$. But in this case the limit would must have domain $\mathbb N$ too, so are we allowed to apply the CLT? After all $N(x,\sigma ^2)$ has domain $\mathbb R$. If you can understand why -- ideally by asking your professor -- it might provide insight into how the CLT works and how this problem can be approached. 
Suppose we have already overcome the problem, in such a way that $S_i$ converge -- in distribution as in the conclusion of the CLT -- to a measurable function $N$ on $\mathbb N$ such that $P(N > b) = 2\varepsilon > 0$. This means that for any fixed $b$ the sequence of real numbers $P(S_i > b)$ converges to $P(N > b)$. 
So for sufficiently high $M$ we have $m > M \implies |P(S_m > b) - P(N > b))| < \varepsilon$. The triangle inequality then gives $P(S_m > b) > \varepsilon$ for all $m \ge M$. So the random variable $\displaystyle \frac{X_1 + X_2 + \ldots + X_m}{\sqrt m} > b$ with nonzero probability. Thus $X_1 + X_2 + \ldots + X_m > b \sqrt{m} > b$ with nonzero probability for each $m > M$. Now what does this mean for the random walk on the whole. Can you express the chance we eventually wander past $b$, in terms of the random variables $X_i$?
