Talk:Random measure

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[edit] Point process and random measure

I would be rather shocked if you could find anywhere in Kallenberg's book a synonymous use of point process and random measure. If I am wrong, could you (or someone) please point out the corresponding passage to me?

I actually doubt any synonymous treatment of the two terms by an authorative source.

Slartibarti 10:56, 25 May 2007 (UTC)

It sure looked that way to me somewhere on the first few pages in the book, a lemma about some kind of equivalence between the two I think. But I do not have the book around any more, had to return it to interlibrary loan. I'll get it again. If you think that the article is wrong, please be bold and edit it to make it right. If you can spend some effort to add more information to help make the topic understandable to mere mortals and maybe even point out what the important results are, that would be particularly nice. This article started from my effort to find out if there are some theorems about random measures that may be useful in the analysis of particle filters and such; but the literature turned out simply impossible to read. Jmath666 08:00, 26 May 2007 (UTC)

I made a few small changes to the article. I know the book by Kallenberg rather well (and have it actually with me at the moment). I think what you are referring to is Lemma 2.3. However, this lemma states that a random measure can be written as the sum of a point process and a so-called non-atomic random measure. This second summand makes a random measure something much more complex than a point process. I plan to add some more contents to this article about general random measures rather than point processes (note that there is already a separate article about point processes, where I have already put quite a bit of effort in, so I think in the long run the point process specific information should disappear from the random measure article. Slartibarti 11:23, 26 May 2007 (UTC)

It's great to have you here. I could not make much sense out of the Kallenberg book; for one thing, there seemed to be a complete lack of motivation and the generality looked rather self-serving. No indication what might be important and useful (and therefore I would consider spending some time on) and what might be done just because it can be done even if it may have no purpose. I liked how you started redoing point processes, and asked a question on the talk page there. Jmath666 06:40, 27 May 2007 (UTC)

Perhaps the statement of point process should stay as an example of what random measures may be for, but specifics should of course go in it's own article. Also you might consider merging - I did notice that in the literature usually "random measures and point processes" is said in one breath. Are there any other examples of random measure than a point process that are useful for something? There are weak solutions of PDEs of course, and if those are stochastic PDEs and the solutions are measures they would be random measures. I do not know if that is relevant here. Jmath666 06:51, 27 May 2007 (UTC)

Thanks for the kind words. I agree that Kallenberg's book is a bit of a shocker, and would certainly not recommend it to anyone who is new to the topic, even if he/she has a very strong mathematical background. It is a really great book, however, (at least for a more theoretical probabilist like me) once you have a general idea of what is going on. I think statements about general random measures are quite often made just because they can be made; which is ok, I think: if you have proved something about point processes and realize that it holds also for general random measures, why not formulate it that way and maybe one day somebody comes along who can use it (or even if not, it might be "just" a beautiful result in itself).

Nevertheless, there are some applications of random measures, although as far as I know much fewer than of point processes. Essentially, random measures can appear if you have a random object "in a random environment". For example: the Cox point process (and I actually rather mean a spatial variant of what is described in the link) is essentially a Poisson process for which the parameter measure is random (i.e. the preference of the points for certain regions is governed by a random environment). Study of the Cox process requires study of the random measure that governs it. Anyway, details would lead to far here. I will explain some of the things when I make changes to the article, which I hope is soon... but time is always a problem, of course. Slartibarti 11:30, 27 May 2007 (UTC)

Thank you. So what would you recommend for someone who is new to the topic, please? Jmath666 15:34, 27 May 2007 (UTC)

Hm, tricky question... I found the first chapter of Karr, Point Processes and their Statistical Interference, 2nd ed., 1991 (ISBN 0-824-78532-0) pretty useful when I first learned about the topic. Daley and Vere-Jones, Introduction to the Theory of Point Processes (ISBN 0-387-96666-8 and ISBN 0-387-95541-0) is surely comprehensive and nice, with a lot of motivation and examples, but sometimes they do things a bit in an unusual way (like the thing with the local compactness). I only know the first edition though and the second edition is massively revised. Then there is Stoyan, Kendall and Mecke, Stochastic Geometry (ISBN 0-471-95099-8), which contains chapters about point processes (nicely written and motivated, leaves out some of the more technical details, but does not "lie"); and also Møller and Waagepetersen, Statistical Inference and Simulation for Spatial Point Processes (ISBN 1-584-88265-4), which I find very pleasant to read (strong bias towards statistics though; does sometimes "lie" a little to simplify things, and there are rather many typos). Slartibarti 02:30, 28 May 2007 (UTC)

Thanks. I'll get Daley and Vere-Jones next trip to the library. Grandell (cited in this article) looks good esp. for convergence of random measures (which is what I need to learn about) except he is doing everything in 1D while my application requires random measures on at least Rⁿ and ideally on a Hilbert space. Jmath666 17:28, 28 May 2007 (UTC)

Random measures are important for stable processes (these are not point processes). A good reference is the book | Stable Non-Gaussian Random Processes" by G. Samorodnitsky and M. Taqqu. Slawekk 04:32, 2 June 2007 (UTC)

I did not find any reference to random measures in the stable processes article. I'll see about the book is (ISBN 0-412-05171-0). Thanks Jmath666 00:39, 4 June 2007 (UTC)

Yes, the article is about stable distributions rather than stochastic processes based on them. I just wanted to indicate in what sense those processes are "stable". Unfortunately, there is no Wikipedia article about stable processes (except for gaussian processes). Note that since Gaussian distribution is a special case of stable distribution, facts about random measures as related to stable processes apply to Gaussian processes as well. I found it very intuitive to think about Gaussian and Poisson processes in terms of intergrals w.r.t. random measures. Slawekk 17:31, 5 June 2007 (UTC)