Let $X_1, X_2, X_3, \ldots$ be i.i.d. random variables with zero mean and let $S_n := X_1 + \ldots + X_n$. Does $T := \inf\{n: S_n > 0\}$ always have infinite first moment?
In the trivial case, where $X_i = 0$, we have $T = \infty$. For random walk, in which $X_i = \pm 1$ with probability $1/2$, it can be shown that $\mathbb{E}[T]=\infty$. I am wondering if it is always the case that $\mathbb{E}[T]=\infty$.
The answer is yes. This follows indirectly from the optional stopping theorem. The relevant formulation of the theorem says the following. If $X_n$ is a martingale, the increments of $X_n$ are "conditionally bounded", and $\mathbb{E}(\tau) < \infty$, then $\mathbb{E}[X_\tau]=\mathbb{E}[X_0]$. "Conditionally bounded" means that $\mathbb{E}(|X_{n+1}-X_n| \, | \mathcal{F}_n) \leq C$ independent of $n$.
Here $S_n$ is a martingale because it is a sum of iid variables and its increments are conditionally bounded by $\mathbb{E}[|X_i|]$. So the first two hypotheses hold. Now by contraposition, since the conclusion fails, the third hypothesis must also fail. That is, since $\mathbb{E}[S_\tau] > 0$ while $\mathbb{E}[S_0] = 0$, we conclude that $\mathbb{E}(\tau)=\infty$.