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Multidimensional Point Process


A multidimensional point process is a measurable function from a probability space (Omega,A,P) into (X,Sigma) where X is the set of all finite or countable subsets of R^d not containing an accumulation point and where Sigma is the sigma-algebra generated over X by the sets

 F_B(k)={X in X:Card(X intersection B)=k}

for all bounded Borel subsets B subset R^d. Here, Card(A) denotes the cardinality or order of the set A.

A multidimensional point process is sometimes abbreviated MPP, though care should be exhibited not to confuse the notion with that of a marked point process.

Despite a number of apparent differences, one can show that multidimensional point processes are a special case of a random closed set on R^d (Baudin 1984).


See also

Marked Point Process, Point Process, Random Closed Set

This entry contributed by Christopher Stover

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References

Baudin, M. "Multidimensional Point Processes and Random Closed Sets." J. Appl. Prob. 21, 173-178, 1984.Matheron, G. Random Sets and Integral Geometry. New York: Wiley, 1975.

Cite this as:

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

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