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

ξ = i I δ X i {\displaystyle \xi =\sum _{i\in I}\delta _{X_{i}}}

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

Examples

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

Uniqueness

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

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

Literature

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