Ah! I see that I’m chasing down the Generalized Central Limit Theorem….

Rabbit holes are fun, but not often conducive to productivity.

]]>It looks like maybe there was a word missing in the paper I quoted (I thought maybe they just needed to drop the -ly). From this paper

it seems that the theorem in Feller says something about all infinitely divisible distributions on the nonnegative integers being compound Poisson, meaning they can be represented as

where is Poisson and are iid and independent of . And infinitely divisible seems to be a condition on the characteristic function of the distribution. It looks like it means that all roots of the characteristic function are themselves characteristic functions.

]]>So, your comment got me to thinking about the relationship between falling powers moments (what seem to be referred to as “factorial moments” elsewhere) and cumulants. And it sent me down a rabbit hole called the Umbral Calculus, which I had never heard of before. If you haven’t either, then the introduction of this survey is nice:

Anyway, I found myself reading this paper by Di Nardo and Senato:

Using the umbral calculus, it details the relationships between moments, factorial moments, central moments, and cumulants of random variables. The Poisson distribution seems to come up a lot. There’s a lot of notation and (unfortunately) I don’t really have the time to understand it fully. But here’s a few tidbits I pulled out that might be relevant to the present discussion:

1. Proposition 7.1 shows that the factorial moment generating function is where is the ordinary moment generating function. So, for the Poisson with mean , we’d get demonstrating the property you used in your proof above, i.e., the the r-th factorial moment of the Poisson is just .

2. Footnote 3 is a quote giving a short history of cumulants, which I found interesting.

3. The first sentence of section 4 says:

The family of Poisson r.v.’s plays a crucial role in the theory of r.v.’s, especially because the most general infinitely [sic] distribution may be represented as a limit of an appropriate sequence of compound Poisson processes (cf. [4]).

Now, [4] is the second volume of Feller’s classic treatise on probability theory. I’m almost ashamed to admit that I don’t have access to a copy of that text, so I’m not really sure what that statement means, but it sounds like it might be relevant to your classical limit, no? After all, you understand the limit for Poisson distributions….

]]>I doubt our paths will ever cross, but I do appreciate the offer and certainly wouldn’t turn down a beer. But, as you noted, this is really just about the fun for me.

]]>If I ever meet you I will gladly buy you a beer or two—or any beverage of your choice. That may recompense you for your unpaid work (though I know you did it just for fun).

I like your approach, because it illustrates a little bit of the power and beauty of cumulants. First, the cumulant generating function of the Poisson distribution is so much simpler than the moment generating function. Second, it’s nice how all the higher cumulants don’t care when you translate the Poisson distribution to make its mean zero. (I bet that’s a general property of cumulants but I’m too lazy to think about it now.)

Your approach is also a bit like my final approach, in the following sense. We avoid working directly with moments and work with other quantities, of which the moments are certain polynomial functions. For me, I happened to notice that when

is a Poisson distribution, these quantities

are simpler than the moments:

They obey

Someone who knew more about Poisson distributions would presumably know all sorts of tricks like this, including cumulants, but I’ve never really studied them before.

]]>If I were smart enough maybe I could do what you’re saying. Can I use some version of the central limit theorem to prove that all the higher centered moments of a Poisson distribution with mean approach certain functions of as ?

I would like to know. But anyway, I got the job done some other way.

]]>off the top of my head, can’t you just invoke the central limit theorem and state that for a large number of particles the Poisson distribution looks like a Gaussian with the correct mean and variance up to and then all moments become “classical” (whatever that means here)?

]]>where the mean is . We have several random variables to worry about. The ordinary number operator has MGF:

The rescaled number operator has MGF:

Finally, we have the -centered, rescaled number operator which has MGF:

Now, the CGF of the -centered, rescaled number operator is

So, the first cumulant of is

where we recall that . For higher order cumulants, the extra term gets killed and we get the same answer as for the rescaled number operator, i.e., for we have that

Thus, all cumulants for the centered, rescaled number operator of order greater than are proportional to and the first cumulant is zero. Furthermore, the moments of the centered, rescaled number operator can be written as polynomials of degree 1 or greater in its cumulants. Therefore, all of the -centered moments of the rescaled number operator are either zero or proportional to and hence go to zero in the limit.

I’ve always been partial to direct demonstration proofs, so I like your proof better. But I’ve already spent so much time on this that I felt I should finish it…. Now I’d better get back to the work I’m being paid for. :)

]]>If a system have a operator distribution that depend on the temperature, then it is possible to apply the statistical mechanic over the second quantization.

If this happen there is a connection between Feynmann diagram and chemical reaction (from elementary particles interaction to molecule chemical reaction potential).

Can the Feynmann diagram be applied to the molecule reaction using simply an approximation of the reaction potential instead of the potential between elementary particles? ]]>