Simple point process
A simple point process is a special type of point process in probability theory. In simple point processes, every point is assigned the weight one.
Definition
Let [math]\displaystyle{ S }[/math] be a locally compact second countable Hausdorff space and let [math]\displaystyle{ \mathcal S }[/math] be its Borel [math]\displaystyle{ \sigma }[/math]-algebra. A point process [math]\displaystyle{ \xi }[/math], interpreted as random measure on [math]\displaystyle{ (S, \mathcal S) }[/math], is called a simple point process if it can be written as
- [math]\displaystyle{ \xi =\sum_{i \in I} \delta_{X_i} }[/math]
for an index set [math]\displaystyle{ I }[/math] and random elements [math]\displaystyle{ X_i }[/math] which are almost everywhere pairwise distinct. Here [math]\displaystyle{ \delta_x }[/math] denotes the Dirac measure on the point [math]\displaystyle{ x }[/math].
Examples
Simple point processes include many important classes of point processes such as Poisson processes, Cox processes and binomial processes.
Uniqueness
If [math]\displaystyle{ \mathcal I }[/math] is a generating ring of [math]\displaystyle{ \mathcal S }[/math] then a simple point process [math]\displaystyle{ \xi }[/math] is uniquely determined by its values on the sets [math]\displaystyle{ U \in \mathcal I }[/math]. This means that two simple point processes [math]\displaystyle{ \xi }[/math] and [math]\displaystyle{ \zeta }[/math] have the same distributions iff
- [math]\displaystyle{ P(\xi(U)=0) = P(\zeta(U)=0) \text{ for all } U \in \mathcal I }[/math]
Literature
- Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
- Daley, D.J.; Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods,. New York: Springer. ISBN 0-387-95541-0.
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