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 $S$ 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
:$\xi =\sum_ \delta_$

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
:$P(\xi(U)=0) = P(\zeta(U)=0) \text U \in \mathcal I$

Literature

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*{{cite book , last1=Daley , first1=D.J. , last2= Vere-Jones , first2= D. , year=2003 , title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods,, location= New York , publisher=Springer , isbn=0-387-95541-0

Point processes