I've come across two statements in a proof that I don't really understand.
Let $X_{i}$ be iid with values in $\{-1,0,1\}$ all with positive probability. Define $S_{n}=\sum_{i=0}^{n}X_{i}$ and the stopping times $\tau_{a}=\inf\{n>0\mid S_{n}=a\}$ and $\tau=\min\{\tau_{a},\tau_{b}\}$.
It is now stated that the stopping time $\tau$ is finite a.s. and that this can be seen by the Law of Large Numbers but I don't see how?
The second statement says that this stopping time $\tau<\infty$ $P_{x}$-a.s. which can be seen by the Central Limit Theorem. But again I don't see how this conclusion is drawn.
Could anyone help me see why this is the case?