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

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Let  S {\displaystyle S} be a locally compact second countable Hausdorff space and let  S {\displaystyle {\mathcal {S}}} be its Borel  σ {\displaystyle \sigma } -algebra. A point process ξ {\displaystyle \xi } , interpreted as random measure on  ( S , S ) {\displaystyle (S,{\mathcal {S}})} , is called a simple point process if it can be written as

ξ = ∑ i ∈ I δ X i {\displaystyle \xi =\sum _{i\in I}\delta _{X_{i}}}

for an index set I {\displaystyle I} and random elements X i {\displaystyle X_{i}} which are almost everywhere pairwise distinct. Here  δ x {\displaystyle \delta _{x}} denotes the Dirac measure on the point  x {\displaystyle x} .

## Examples

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Simple point processes include many important classes of point processes such as Poisson processes, Cox processes and binomial processes.

## Uniqueness

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If  I {\displaystyle {\mathcal {I}}} is a generating ring of  S {\displaystyle {\mathcal {S}}} then a simple point process  ξ {\displaystyle \xi } is uniquely determined by its values on the sets  U ∈ I {\displaystyle U\in {\mathcal {I}}} . This means that two simple point processes  ξ {\displaystyle \xi } and  ζ {\displaystyle \zeta } have the same distributions iff

P ( ξ ( U ) = 0 ) = P ( ζ ( U ) = 0 )  for all  U ∈ I {\displaystyle P(\xi (U)=0)=P(\zeta (U)=0){\text{ for all }}U\in {\mathcal {I}}}

## Literature

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* Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77\. 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.