Talk:Compound Poisson process

Please. It does not make sense to say "Let ... be a compound Poisson process... Then ...." when you're DEFINING the concept of compound Poisson process. Only when stating a THEOREM or proposition about such a process does that syntax make any sensse. Michael Hardy 22:49, 30 Dec 2004 (UTC)

Merging with Poisson process

Disagree about merging this article with Poisson process. That one is already a big article. Rather, this article should be expanded. Oleg Alexandrov 01:29, 1 Jun 2005 (UTC).

I absolutely agree with Oleg. PH Potgieter Sep 24 11:13:35 SAST 2005.

So do I. R Binter

I added a bit to the article and i really don't think it should be merged, would anyone be opposed to me removing the merge request. Faisel Gulamhussein 20:42, 6 January 2006 (UTC)

I'd rather propose to merge this article with the article on the Compound Poisson Distribution. Both articles merely differ by time dependence of the CCP. Syngola (talk) 14:17, 19 August 2014 (UTC)

Is the variance correct?

I am a common user of wikipedia.org. There is an entry "compound Poisson process), on which page the mean value and the variance are given.

However, I can get the mean value but cannot the variance as shown on the page.

I want to make sure whether the variance value calculation on that page is correct.

Thanks a lot.

(The above is the text of an email I got today. Oleg Alexandrov (talk) 17:24, 24 May 2006 (UTC))

If the definition of the process is correct, then the mean and variance look right to me. Also, was the person who sent you the mail the one who made the anonymous edits earlier today? Lunch 19:08, 24 May 2006 (UTC)

I don't know, could be. Oleg Alexandrov (talk) 19:57, 24 May 2006 (UTC)

I added final calculation of the moment generating function, and it confirms the result. I didn't check the individual steps in the variance calculation, though. — Arthur Rubin | (talk) 20:12, 24 May 2006 (UTC)

Using "well-known" properties of the moment generating function, $\operatorname {var} (X)=\left.{{\frac {d^{2}}{ds^{2}}}\ln M_{X}(s)}\right|_{s=0}$ and $E\left(X^{2}\right)=\left.{\frac {d^{2}}{ds^{2}}}M_{X}(s)\right|_{s=0}$, we can easily see the desired result. — Arthur Rubin | (talk) 20:22, 24 May 2006 (UTC)

I've just gone through the derivation of the variance and it certainly looks correct to me. To address the concern of your anonymous email correspondent, I'd need to know more specifically which parts he or she does not understand. Michael Hardy 21:06, 24 May 2006 (UTC)

semi-martingale representation

How to represent compound Poisson process in a semi-martingale form? And is this representation unique?

Is the rate λ still correct for the compounded poisson process?

I think the rate of the compounded poisson process has to be λ*f(x) where f(x) is the density of the iid random variables

using the notation from the article, note that the expected value of a compound poisson process is $E(Y(t))=\lambda tE(D)$ (versus λt for a simple poisson process). is this what you're referring to? Lunch 21:41, 1 June 2006 (UTC)

Depends on what you mean by "rate". Michael Hardy 21:52, 1 June 2006 (UTC)

I mean the intensity. I'm interested in the intensity of a Compound Poisson Process where the Poisson process has intensity λ.

What about zero?

Forgive me if this is a stupid question but pretty much the first line defines the sum as being indexed by i from 1 to N(t) but then allows i to be 0 in some cases, which is surely natural within the Poisson distribution. Should the sum then be changed to allow for the atom of probability at 0? Or is there something I'm missing? {subst:unsigned2|13:53, November 12, 2006|89.124.143.218}}

I'm afraid it is a stupid question. Having the sum start at one allows for the atom of probability at zero where N(t)=0. — Arthur Rubin | (talk) 16:26, 12 November 2006 (UTC)

Surely this now leads to a bit of an ambiguity. If $N(t)=0$ then (taken from the entry on summation) there are no terms to sum and $Y(t)=0$. In which case why define $D_{0}$? —The preceding unsigned comment was added by 134.226.1.194 (talk • contribs) 09:32, November 13, 2006 (UTC)

That's a good point. D₀ gone. — Arthur Rubin | (talk) 14:36, 13 November 2006 (UTC)

Existence of the mean?

I feel it might be appropriate to add the assumption $E[|D_{1}|]<\infty$ before calculating the mean of the process.

Fladnaese (talk) 18:47, 26 May 2011 (UTC)

Simar's algorithm might not the right choice here

Simar's algorithm, despite the title of the paper, is for mixtures of Poisson distributions, and not for the compound Poisson process as referred to in this article. — Preceding unsigned comment added by 193.52.24.78 (talk) 11:17, 30 August 2013 (UTC)