Talk:Discrete-stable distribution

Limit theorems

Something must be missing from the description of limit theorems, since the sum of many discrete iid random variables converges to a normal distribution, which is not a discrete-stable distribution. Rlendog (talk) 15:26, 28 March 2015 (UTC)

I have tried to make it more explicit that in the case when the distribution is continuous and has finite variance, the limit is the normal distribution as you point out. In the case that the distribution is discrete has finite mean, the limit is the Poisson distribution. Wainson (talk) 20:01, 16 April 2015 (UTC)

That is not the case. Under the Central Limit Theorem, the normal distribution is also the limit a discrete distribution with finite variance. The CLT is not dependent on discrete vs. continuous. A binomial and even Poisson itself tends to the normal. Rlendog (talk) 02:31, 17 April 2015 (UTC)

We have illustrations of this here and here. Rlendog (talk) 02:34, 17 April 2015 (UTC)

I removed the section on limit theorems. I am not finding any support for discrete stable distributions satisfying this property except for lattice random variables. In fact [1] explicitly states that at least some discrete stable distributions belong "to the domain of normal attraction of the absolutely continuous positive stable distribution (stable distribution with index of stability α = γ and with skewness parameter β = 1), whose Laplace transform is given by exp(−λtγ )." (emphasis added) Rlendog (talk) 03:09, 17 April 2015 (UTC)

Some of this confusion is perhaps due to the fact that in the limit one can approach, in distribution, both continuous and discrete RVs; Indeed a sum of independent Poisson RVs is Poisson distributed, and approaches a normal distribution in the limit, in a certain sense. Perhaps we should mention the discrete limit theorems are special cases of the continuous ones? Some care is needed to indicate which limits we are taking, and what kind of convergence. Random sums of Poisson RVs can themselves approach non-Poisson Stable RVs in the limit, according to the linked reference from Lee, for example. Maybe here's not the case to explain that? In whcih case we should remove the section entirely again; currently it's ambiguous what sort of limits we are discussing etc. --Livingthingdan (talk) 08:11, 6 December 2015 (UTC)

• So now I understand better what the article is getting it. The deleted section was not wrong, but there was very important context missing from the article, hopefully I will get around to fixing it.

Briefly: it is usually left unsaid, but actually, when we invoke the central limit theorem what we really mean is a sum and re-scale. The unscaled sum of iid variables doesn't converge to anything at all! Of course, if you take a distribution with support on the non-negative integers, and multiply it by a non-integer scalar, the resulting distribution will not be supported by the non-negative integers. But there is an alternate re-scaling operation which does yield an integer-supported distribution, and that has a central limit theorem that converges to a Poisson distribution (or more generally, a discrete-stable distribution, if the finite variance assumption is dropped). --God made the integers (talk) 21:58, 23 December 2016 (UTC)

Generating probability Generating Function definition

We link to a PGF definition in terms of $Es^{X}$ for a RV X, but then work in terms of $E(1-s)^{X}$. We should either switch notation, or clarify.

--Livingthingdan (talk) 08:00, 6 December 2015 (UTC)

What?

The discrete-stable distributions are a class of probability distributions with the property that the sum of several random variables from such a distribution is distributed according to the same family....

The most well-known discrete stable distribution is the Poisson distribution which is a special case as the only discrete-stable distribution for which the mean and all higher-order moments are finite.

The difference of two independent Poisson variables is discrete, stable, and has all moments finite, but it is not Poisson.

Am I misunderstanding "stable" in this context? Or is there supposed to be a difference between "discrete stable" and "discrete-stable"? Or am I missing something else? Help? --God made the integers (talk) 18:54, 14 December 2016 (UTC)

As I indicate above, there is important context missing from the article. --God made the integers (talk) 21:58, 23 December 2016 (UTC)