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 $\mathbb{N}$?

  • 1
    $\begingroup$ I don't think this answers your question but it might be helpful to point out that Nolan has the first chapter of his book on Levy-alpha stable distributions online here: academic2.american.edu/~jpnolan/stable/chap1.pdf . Perhaps you know this already but the Cauchy distribution is a Levy-stable distribution with $\alpha=1$ and (in Nolan's notation) skew parameter $\beta=0$. Setting the skew parameter $\beta=1$ makes it essentially zero to left of a certain point and one can adjust the shift, $\delta$ (again, in Nolan's notation), to taste. $\endgroup$ – dorkusmonkey Nov 4 '11 at 21:14

I realise you asked this question over two years ago, but: The one-sided continuous-stable distributions can be directly linked to the discrete-stable distributions through a formulation called the 'Poisson transform', where the mean of a Poisson distribution is modulated by another distribution (one example of this is where the other distribution is a Gamma distribution - the resultant distribution is negative binomial).

In the case when the modulating distribution is one-sided continuous-stable of power-law index nu, the resultant distribution is discrete-stable with index nu.

I talk about this in depth in my thesis ("Continuous and discrete properties of stochastic processes") which also has an in-depth literature review you might find useful and, for a tl;dr version, in the paper "Continuous and discrete stable processes" (on Academia.edu or DOI link to APS.org).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.