Let $x_1,x_2,...$ be i.i.d real-valued random variables with $\mathbb{E}[x_i] > 0$. Let $S_n = \sum\limits_{i=1}^{n}x_i$ be the partial sums of the r.v.s let and $N= \inf\{ n \mid S_n > h\}$ be the first time the sum crosses the positive threshold $h$. My question has two related parts.
I can show that if the $x_i$ has bounded variance then $P(N < \infty) = 1$. However, it is intuitive to me that this should be true even if the second moment of the random variable does not exist. So how can I prove that $P(N< \infty)$ just assuming $\mathbb{E}[|x_i|] < \infty$? (of course still assuming positive drift)
I have seen it used in many papers that $\mathbb{E}[N] < \infty$. However, I can find no proof of this in any books that I looked at. I have seen proofs that make the additional assumption of a lower barrier, but they do not apply in my case (technically my lower barrier is $-\infty$). How to prove that the expected time is finite?
By the strong law of large numbers $S_n\to \infty$ a.s. Thus, $\mathsf{P}(N<\infty)=1$ (for $h>0$). As for the expectation, for any $r\ge 1$, $\mathsf{E}N^r<\infty$ iff $\mathsf{E}|x_1^{-}|^r<\infty$ (see this paper for details).