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   be a locally compact second countable Hausdorff space and let   be its Borel  -algebra. A point process  , interpreted as random measure on  , is called a simple point process if it can be written as

 

for an index set   and random elements   which are almost everywhere pairwise distinct. Here   denotes the Dirac measure on the point  .

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   is a generating ring of   then a simple point process   is uniquely determined by its values on the sets  . This means that two simple point processes   and   have the same distributions iff

 

Literature

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  • 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.