Suppose we have a random walk on the real line whose step sizes have finite variance. We know that, when viewed as a function and suitably rescaled, this random walk will converge to a Brownian motion. This is Donsker's theorem.
Soft question: what happens when the step sizes have infinite mean? Is it also possible to rescale these infinite-mean random walks in a way such that their measure converges to a tight law in an appropriate function space (not necessarily $C[0,1]$)?
A quick Google search got me to Levy processes, but I had trouble making the connection with random walks. I could just be thick, though. Can anyone point me to a reference which deals with this type of situation?