Consider the iid random variables $X_1,X_2,...$ with $\mathbb{E}[X_i] = \mu$. Let $S_n = X_1+...+X_n$ be a random walk (it could be symmetric/unsymmetric in 1D or higher dimension, but I use the notation for 1D). We know $S_n-\mu n$ is a martingale with respect to the natural filtration. Let $T$ be a hitting time, i.e., $T=\inf\{n\ge0 : S_n = a \}$ for some $a$. I have seen in several textbooks that we can apply the optional stopping theorem to this martingale $M_n = S_n-\mu n$. However, I do not get why the requirements of optional stopping theorem in this case are satisfied. For example, is it true that the stopping time $T$ defined above is almost surely bounded?
P.S. I think the hitting time is finite because random walk is recurrent. Is this true? I wonder if we can conclude that $T$ is almost surely bounded using this fact.