Talk:Point process

extended

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 $X_{i}$ 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)

Local compactness is introduced because it allows to represent a point process as a random variables taking values in the space of Radon measures. Without local compactness, the realisations of a point process are not necessarily Radon measures and there is no natural structure on the resulting space of measures.

constructions

Can any experts provide more constructions for point processes,
especially stationary ones on the entire line or plane ?

A construction for the real half-line is already given.
Is there any way to take 2 half-lines, with a Poisson process on each,
and somehow 'glue' them together to get one process on the whole line ?
The tricky part is the interval between the first points in each half-line.
If there's a way to do this gluing, I'd like to see it.

And what about the entire plane ?
Can you take the product of 2 independent Poisson processes on the entire line ?

--Kanyonman (talk) 13:10, 21 October 2008 (UTC)

Point process as a stochastic process

A point process is a stochastic process indexed by a sigma algebra of sets instead of a subset of the real line (time). But it is still a stochastic process in the general sense of the word. See, e.g., "Diffusions, Markov Processes and Martingales, Vol. 1" by Rogers and Williams, section II.33-37. There they treat Poisson point processes analogously to Brownian motion with a different indexing set. The Daniel-Kolmogorov theorem is applied to show existence of pre-Poisson set functions. Just as Brownian motion has extra properties that need to be satisfied (being a continuous function instead of just a function), the Poisson measure must also satisfy extra properties (being a measure instead of just a set function). — Preceding unsigned comment added by Vinzklorthos (talk • contribs) 15:34, 15 November 2011 (UTC)

Usability issues with this page

I'm sure this page is very nice and technically accurate and all, but it's couched in such abstract language as to be almost totally impenetrable by anyone who just wants a simple introduction to the concept as it might be used in an ordinary engineering or science context. As such a user, I'm not qualified to actually fix this page myself, but to whomever is qualified to edit this page, please have some consideration for those of us who are not professional mathematicians! 69.205.70.143 (talk) 21:21, 29 July 2012 (UTC)

I agree; I am a professional mathematician and I find that this article is poorly written, too abstract to be useful for students or scientists. The definition should start with the case of point processes with values in some finite dimensional space and many examples.

Discrete Time vs. Continuous Time Point Processes

My reading of this is that a point process is defined over continuous time. Is that right? Mrdthree (talk) 03:27, 9 August 2013 (UTC)

opaque

Please can an expert edit the definition, in particular the notation and background around "that renders all the point counts" This is opaque Mlnsmonster (talk) 11:33, 1 July 2014 (UTC)

Is the sum necesarily finite in the representation?

It is written:

"Every point process ξ can be represented as

$\xi =\sum _{i=1}^{N}\delta _{X_{i}},$"

From the definition I don't see why it should be a finite sum. Bongilles (talk) 14:16, 15 October 2013 (UTC)

Why talk about "isolated points" in the beginning and then define simple processes? If the process consists of isolated points a.s., then clearly it is simple, so starting with "isolated points" only would mean that the entire article is only about simple processes. — Preceding unsigned comment added by 78.97.140.148 (talk) 17:48, 21 December 2014 (UTC)

missing coverage of basic introductory topics

Article makes no mention of invariant measure which is what point processes settle down into. No mention of chaos, many important processes are chaotic. No mention of shift spaces, which is the space in which point processes act. No mention of either transfer operators or Koopman operators, which describe the time evolution of point processes, when these are viewed as probability distributions. No examples of basic processes, e.g. the Bernoulli process. This is all standard introductory-textbook material for this topic. 67.198.37.16 (talk) 23:22, 11 May 2018 (UTC)

Wrong link for "counting measure"

In the section "General point process theory", the words "counting measure" link to Counting_measure which seems be accurate at first glance but I think the article Counting_measure describes a different concept (namely the measure that maps a set to its cardinality). Counting_measure defines *the* counting measure which is a specific measure. While on the present page, we talk about "the set of locally finite counting measures" which does not make sense if there is only one. I guess on the present page, *a* counting measure is any integer-valued measure (possibly subject to some restrictions). So the link to Counting_measure should be removed and a definition of "counting measure" added in the text (or the link could lead to a page describing counting measures in the sense of the present page). (I am not sufficiently familiar with point processes to do so myself.) Dominique Unruh (talk) 13:17, 28 September 2019 (UTC)