#+TITLE:Point Processes
#+AUTHOR:Darlan Cavalcante Moreira
#+EMAIL:darcamo@gmail.com
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Consider the \(d\) dimensional Euclidean space \(\bbR^d\). A spacial point
process (p.p.) \(\Phi\) is a random, finite or countably-infinity
collection of points in the space \(\bbR^d\), without accumulation points.


One can consider any given realization \(\phi\) of a point process as a
discrete subset \(\phi = \{x_i\} \subset \bbR^d\) of the space. It is often
more convenient to think of \(\phi\) as a /counting measure/ or a /point
measure/ \(\phi = \sum_{i}\delta_{x_i}\), where \(\delta_{x}\) is the
/Dirac measure/ at \(x\)[fn:1].


More specifically, for a set \(\bbA \subset \bbR^d\) the value of
\(\phi(\bbA)\) will be the number of "points" of \(\phi\) in \(\bbA\).


* Examples

\(\mathbf{y} = c \mathbf{x}\)

\(\Vt{x} = 2 * 3\)
  

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* Other definitions
  - *simple*: point process is known as simple if XXXXXXXXXXXX
  - *expectation measure*: 
  - *Laplace functional*: 
  - *moment measure*:
  - *Stationarity*: For a stationary point process, the mean measure 
    \(E\{\phi (\cdot)\} = \lambda \|\cdot\|\) for some constant \(\lambda \geq 0\)
    and where \(\|\cdot\|\) stands for the *Lebesgue measure*[fn:2]. This
    \(\lambda\) is called the *intensity* of the point process. A
    stationary point process on \(\bbR^d\) has almost surely either 0 or an
    infinite number of points in total.



* Footnotes

[fn:1] If \(\delta_{x}\) is the Diract measure at \(x\) then \(\delta_{x}(\bbA) = 1\) if \(x \in \bbA\) and \(\delta_{x}(\bbA) = 0\) if \(x \notin \bbA\).

[fn:2] The Lebesgue measure is the standard way of assigning a measure to
subsets of n-dimensional Euclidean space. For \(n = 1, 2\), or \(3\), it coincides
with the standard measure of length, area, or volume. In general, it is
also called \(n\)-dimensional volume, n-volume, or simply volume.[1]



Add draft post (in org-mode) about point processes
Add draft post (in org-mode) about point processes
