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

Question

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?

Answer 1

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.

I can't look up a precise statement at the moment, but hopefully this is a start.

Answer 2

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.