The generalized central limit theorem of Gnedenko-Levy describes the asymptotic behavior of a sum of IIDRVs which may not have finite mean or variance. Only a small class of limit laws can be realized, and Cauchy distributions are a notable example.

Steutel and van Harn introduced a notion of stability for RVs defined on $\mathbb{N}$: similarly to the "traditional" case (where characteristic functions are extensively utilized), this is done in terms of generating functions.

In my cursory survey, I have found oblique remarks to the effect that discrete stable distributions are very similar to their continuous analogues. I have, however, been unable to find a usable reference.

In my work I have recently come across a rather surprising bit of behavior. I have some RVs on $\mathbb{N}$ that (after a bit of reasonable kernel density estimation) are profoundly good fits to (discrete analogues of) Cauchy distributions (in fact, up to single-parameter scaling, the same distribution--which is remarkable because their peaks are not at the origin and rescale to each other). This is also pretty weird because--again--these RVs are defined on $\mathbb{N}$, not $\mathbb{Z}$. If the fits weren't so good, I'd toss it off as a coincidence, but I am (I think naturally) curious about the possibility that this is no accident. If the discrete stable distributions are close in a suitable sense to continuous analogues, I would have the seed of an explanation. So:

How similar are discrete and continuous stable distributions, and in particular is there something like a Cauchy distribution centered away from 0 that can occur as a discrete stable distribution supported on $\mathbb{N}$?