simple point process

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

:$\backslash xi\; =\backslash sum\_\{i\; \backslash in\; I\}\; \backslash delta\_\{X\_i\}$

for an index set $I$ and random elements $X\_i$ which are almost everywhere pairwise distinct. Here $\backslash delta\_x$ denotes the Dirac measure on the point $x$.

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

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

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

• {{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|series=Probability Theory and Stochastic Modelling |volume=77 |location= Switzerland |publisher=Springer |doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3}}
• {{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}}

Category:Point processes