TOPICS
Search

Stationary Point Process

A stationary point process is a point process whose statistical distribution is invariant under translations. On R^d, this means that for every positive integer r and all bounded Borel subsets A_1, ..., A_r, the joint distribution of (N(A_1+t),...,N(A_r+t)) does not depend on the translation t in R^d (Daley and Vere-Jones 2007).

On the real line, a weaker condition is that, for every x>0 and every nonnegative integer k,

Pr{N(t,t+x]=k}

depends on x but not on t. This condition is called crude stationarity. It specifies single-interval count distributions without requiring invariance of their joint distributions. A related condition is interval stationarity.

When the intensity function rho is constant, the point process is called first-order stationary or homogeneous (Pawlas 2023). Constant intensity function alone does not imply stationarity of the full statistical distribution.

See also

Bond Percolation, Bootstrap Percolation, Cayley Tree, Cluster, Cluster Perimeter, Lattice Animal, Percolation Theory, Percolation Threshold, Polyomino, Random Walk, s-Cluster, s-Run, Site Percolation

Explore with Wolfram|Alpha

References

Daley, D. J. and Vere-Jones, D. An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods, 2nd ed. New York: Springer, 2003.Daley, D. J. and Vere-Jones, D. An Introduction to the Theory of Point Processes Volume II: General Theory and Structure, 2nd ed. New York: Springer, 2007.Pawlas, Z. "Spatial Modeling and Spatial Statistics." 2008. http://www.math.ku.dk/~pawlas/rumlig.pdf.Pawlas, Z. "Spatial Modelling (NMTP438)." 2023. https://www.karlin.mff.cuni.cz/~pawlas/2023/MTP438/spatial.pdf.

Referenced on Wolfram|Alpha

Stationary Point Process

Cite this as:

Weisstein, Eric W. "Stationary Point Process." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StationaryPointProcess.html

Subject classifications