Proving that a random walk that diverges to infinity may not become negative Consider a random walk $S_n= \sum_{k=1}^n X_k$, where $\{X_k\}_{k=1}^\infty$ are independent and identically distributed random variables. Assume that $S_n \rightarrow \infty$ almost surely as $n \rightarrow \infty$. Let $$\tau = \inf\{n \geq 1: S_n \leq 0\}.$$
In the book "Stopped Random Walks - Limit Theorems and Applications" by Allan Gut I found a theorem stating that under the assumptions above $\tau$ is defective, i.e., $$\mathbb{P}(\tau = \infty)>0.$$ However, no proof for the theorem is provided. Could anyone provide any hints on how to prove this theorem? Also, does anyone know if there are any generalizations of the theorem for the case that $\{X_k\}_{k=1}^\infty$ are not i.i.d.? Thank you for your time!
 A: The random walk $(S_n; n \ge 0)$ satisfies the strong Markov property.  In more detail, the random variable $\tau$ is a stopping time, i.e. adapted to the process, so the strong Markov property implies the trajectory $(S_{\tau+n}; n \ge 1)$ after $\tau$ is independent of $(S_k; 0\le k < \tau)$ given $S_\tau$.  Also, the random walk is homogeneous in time and space so the distribution of the trajectory after $\tau$ is also the same.
Suppose that $\mathbb{P}(\tau = \infty) = 0$, i.e. $\mathbb{P}(\tau < \infty) = 1$ and the Markov chain a.s. goes $\le 0$ in finite time.  The idea is to use this and the SMP to show the chain repeatedly goes $\le 0$ and thus $S_N \not\to \infty$.
Let $\tau_0 = \tau$ and recursively define $\tau_{n+1} = \inf\{\tau > \tau_n: S_\tau \le S_{\tau_n}\}$.  Note that by construction $0 \ge S_{\tau_0} \ge S_{\tau_1} \ge \ldots$.  Moreover, $\tau_0$ is a.s. finite by supposition.  The SMP implies $(S_{\tau_0 + n}; n \ge 1)$ is independent of the past (given $S_{\tau_0}$, and the precise value $\le 0$ doesn't matter) and homogeneity of the random walk implies $\tau_1$ has the same distribution* as $\tau_0$ and in particular is finite a.s.  Applying the argument repeatedly shows $\tau_n$ is finite a.s.  So $\tau_n \to \infty$ and so $S_{\tau_n} \le 0$.  Hence $S_n \not\to \infty$.
Regarding relaxed assumptions- Do you have a specific scenario in mind for the non-i.i.d case?  (You can prove something along the lines of my answer for time homogenous Markov chains with extra conditions when starting from $\le 0$ at step *)
