Simple point process

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

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

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

Examples

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

Uniqueness

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

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

Literature

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.