# Does a random walk with infinite mean ever converge to anything?

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?

I think what you're looking for is the so-called "stable FCLT", stating that such a random walk, appropriately rescaled, converges weakly in Skorohod space to $(\alpha, \beta)$-stable Levy motion. It looks like this is due to Skorohod and the canonical modern reference is Jacod and Shiryaev.
You might also look at Kallenberg's book Foundations of Modern Probability Theory (specifically Theorem 16.14 in the second edition). A simple example: the random walk with step distribution the standard Cauchy distribution, which is symmetric about $0$ but has infinite mean. This random walk (scaled in space by $1/n$ rather than by $1/\sqrt{n}$) converges in law (in the space of right-continuous, left-limited paths) to the standard Cauchy process.