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.

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

${\displaystyle \xi =\sum _{i\in I}\delta _{X_{i}}}${\displaystyle \xi =\sum _{i\in I}\delta _{X_{i}}}

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

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

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

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

• 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.

• Point processes