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Simple Point Process


A simple point process (or SPP) is an almost surely increasing sequence of strictly positive, possibly infinite random variables which are strictly increasing as long as they are finite and whose almost sure limit is infty. Symbolically, then, an SPP is a sequence T=(T_n)_(n>=1) of R^^_0-valued random variables defined on a probability space (Omega,F,P) such that

1. P(0<T_1<=T_2<=...)=1,

2. P(T_n<T_(n+1),T_n<infty)=P(T_n<infty),

3. P(lim_(n->infty)T_n=infty)=1.

Here, R^^_0=[0,infty] and for each n, T_n can be interpreted as either the time point at which the nth recording of an event takes place or as an indication that fewer than n events occurred altogether if T_n<infty or if T_n=infty, respectively (Jacobsen 2006).


See also

Marked Point Process, Point Process, Self-Correcting Point Process, Self-Exciting Point Process, Spatial Point Process, Spatial-Temporal Point Process, Temporal Point Process

This entry contributed by Christopher Stover

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References

Jacobsen, M. Point Process Theory and Applications: Marked Point and Piecewise Deterministic Process. Boston: Birkhäuser, 2006.

Cite this as:

Stover, Christopher. "Simple Point Process." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SimplePointProcess.html

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