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I am trying to understand the definition of point process when reading its Wikipedia article:

Let $S$ be locally compact second countable Hausdorff space equipped with its Borel σ-algebra $B(S)$. Write $\mathfrak{N}$ for the set of locally finite counting measures on $S$ and $\mathcal{N}$ for the smallest σ-algebra on $\mathfrak{N}$ that renders all the point counts

$$ \Phi_B : \mathfrak{N} \to \mathbb{Z}_{+}, \varrho \mapsto \varrho(B)$$

for relatively compact sets $B$ in $B$-measurable.

A point process on $S$ is a measurable map $ \xi: \Omega \to \mathfrak{N} $ from a probability space $(\Omega, \mathcal F, P)$ to the measurable space $(\mathfrak{N},\mathcal{N})$.

My questions are:

  1. Is the counting measure the one that gives the cardinality of a measurable subset, as defined in its Wikipedia article? If yes, isn't it that there is only one counting measure on a measurable space, and why in the definition of point process, does "write $\mathfrak{N}$ for the set of locally finite counting measures on $S$" imply that there are more than one counting measures on $S$?
  2. It has been noted[citation needed] that the term point process is not a very good one if S is not a subset of the real line, as it might suggest that ξ is a stochastic process.

    Is a point process a stochastic process?

    If no, when can it be? How are the two related?

Thanks and regards!

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1 Answer

up vote 2 down vote accepted

(1) Every locally finite counting measure on $S$ is of the form $\sum_{x\in \Lambda}\delta_x$ where $\Lambda$ is a locally finite subset of $S$. That is, $\Lambda$ should intersect every compact set only finitely often. Of course, there are infinitely many choices of subset $\Lambda$, and thus plenty of counting measures.

(2) In the case where $S=[0,\infty)$ we can define an integer valued stochastic process by setting $X(t)=\xi[0,t]$. That is, $X(t)$ is the amount of mass that the (random) measure $\xi$ assigns to the set $[0,t]$. For instance, the Poisson process can be expressed in this way. But for a general point process, there may be no notion of a "time parameter" and so it is not thought of as a stochastic process.

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Thanks! I am still wondering about the defintion of "a locally finite subset of S". For better understanding, can "Λ should intersect every compact set only finitely often" be more mathematically rewritten? Must $\Lambda$ be a countable set, so that the sum $\sum_{x\in \Lambda}\delta_x$ can make sense? –  Tim Apr 20 '11 at 2:26
    
Yes, since $S$ is $\sigma$-compact every locally finite set $\Lambda$ is countable. But not every countable set is locally finite: on the real line compare $\lbrace n: n \geq 1\rbrace$ and $\lbrace 1/n :n \geq 1\rbrace$. –  Byron Schmuland Apr 20 '11 at 2:32
    
Thanks! How to understand "Λ should intersect every compact set only finitely often"? –  Tim Apr 20 '11 at 2:37
    
@Tim I don't know if there is anything better than that sentence. In papers I have seen it expressed as "$|\Lambda\cap K|<\infty$ for every compact set $K$" where $|\ \cdot\ |$ means "cardinality". –  Byron Schmuland Apr 20 '11 at 2:39
    
Thanks! (1) Is a point process a stochastic process, if and only if the locally finite counting measure on S all have distribution functions? (2) For a point process $\xi$, is $\xi(A), \forall A \in B(S)$ always a measurable mapping, i.e. random variable? (3) Are there any references regarding my two questions? Thanks! –  Tim Apr 28 '11 at 4:59
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