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 $𝒮$ be its Borel $\sigma$-algebra. A point process $\xi$, interpreted as random measure on $(S,𝒮)$, 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 $ℐ$ is a generating ring of $𝒮$ then a simple point process $\xi$ is uniquely determined by its values on the sets $U\in ℐ$. This means that two simple point processes $\xi$ and $\zeta$ have the same distributions iff

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