Talk:Point process

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I extended the article, so that it covers what are in my eyes the three major aspects of point processes in current mathematics (1. the theoretical approach, 2. point processes in spatial statistics, 3. classical point processes on the real (half-)line). I have tried to delete none of the old information, but have moved most of it to the appropriate section and sometimes reformulated it in terms of more general concepts.

There are still many things to be done to make the information that is here up to now more understandable (especially: please do correct my English). My main plan is to add some examples and pictures, especially to the Section "Point processes in spatial statistics".

Slartibarti 10:46, 26 May 2007 (UTC)

I am having a bit hard time make some sense out of the definition. So, according to the definion, if Xi are V valued random variables, where V is not locally compact - say an infinitely dimensional Banach space - then
\xi=\sum_{i=1}^N \delta_{X_i}
is not a point process? Why, would some fundamental theorems not hold any more? What are the fundamental theorems here, please? Any help in making it understandable will be much appreciated. Jmath666 06:22, 27 May 2007 (UTC)

Local compactness is usually assumed in the literature, but I think you get quite far without it. The book by Daley and Vere-Jones (at least the older edition that I cite in the article) defines point processes on (not necessarily locally compact) complete separable metric spaces, which is in a sense more general, because our space in the article is Polish. I think they mention something about being somewhat "unconventional" in this respect in the introduction, but I don't have the book with me at the moment. Anyway, there is the book by Kallenberg, and there are the books by Karr (Point Processes and their Statistical Inference), and Stoyan, Kendall, and Mecke (Stochastic Geometry and Its Applications), which I am sure use all locally compact second countable Hausdorff spaces.
I am not quite sure right now what exactly would go wrong without local compactness. One problem that I see is that the "realization point patterns" could have limit points (namely in any point that does not have a compact neighborhood) if we continue to require only that they are locally finite (and if we require something stronger, then that would probably depend on the concrete metric, which is a bit unesthetic).
Slartibarti 10:55, 27 May 2007 (UTC)