I am seeking conditions on the distribution of the step sizes that guarantee that a random walk on the 2D lattice will return to the origin (with probability 1). Essentially, under what conditions can Pólya's theorem be proved? Certainly it holds if the steps are of size 1 (and equally probable in all directions), and I believe if the variance of the step sizes is finite, the return probability is still 1. But what about distributions with infinite variance, like the Lévy distribution? Does "Lévy flight" return to the origin? And more generally, are there conditions on the distribution that guarantee this return?
This is likely all well known, and if so, pointers to the relevant literature would be appreciated. Thanks!