Assume $\xi_1, \xi_2, \dots \in \mathbb{Z}$ are i.i.d. random variables, and $\mathbb{E}\xi_i$ is undefined (that is, $\mathbb{E}\xi_i^+=+\infty$ and $\mathbb{E}\xi_i^-=-\infty$). Let $S_0=0, S_n=\xi_1+\xi_2+\dots+\xi_n$. Is it possible to prove that random walk $S_n$ is recurrent or transient (i.e. returns to the origin infinitely or not)?
I've seen a similar question here, and I've tried to work out the infinite expectation case myself. But I don't know how to deal with the undefined expectation case. I've tried to prove something like $\mathrm{P}(\bigcap\limits_{n=k}^{+\infty}\{|S_n|>M\})>0$, but to no avail.