Point process notation: Difference between revisions

Revision as of 10:51, 17 October 2013

Revision as of 15:07, 31 October 2013

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:<math> N(B) =\#( B \cap \Phi), </math>

where <math> N(B)</math> is a [[random variable]] and <math> \#</math> is a [[counting measure]], which gives the number of points in some set. In this [[mathematical expression]] the point process is denoted by <math> \Phi</math> while <math> N</math> represents the number of points of <math> \Phi</math> in <math> B</math>. In the context of random measures, one can write <math> N(B)=n</math> to denote that there is the set <math> B</math> that contains <math> n</math> points of <math> \Phi</math>. In other words, <math> N</math> can be considered as a [[random measure]] that assigns some non-negative integer-valued [[Measure (mathematics)|measure]] to sets.<ref name="stoyan1995stochastic"/> This interpretation has motivated a point process being considered just another name for a ''random counting measure''<ref name="molvcanov2005theory">I.{{cite Sdoi|10. Mol1007/1-84628-150-4}}</ref>{\vc{rp|106}anov} and the techniques of random measure theory offering another way to study point processes,<ref name="stoyan1995stochastic"/><ref name="grandell1977point">J. Grandell. Point processes and random measures. ''TheoryAdvances ofin randomApplied setsProbability''., pages Springer502–526, 20051977.

</ref> and the techniques of random measure theory offering another way to study point processes,<ref name="stoyan1995stochastic"/><ref name="grandell1977point">J. Grandell. Point processes and random measures. ''Advances in Applied Probability'', pages 502–526, 1977.

</ref> which also induces the use of the various notations used in [[Integral#Terminology and notation|integration]] and measure theory. {{efn|As discussed in Chapter 1 of Stoyan, Kendall and Mechke,<ref name="stoyan1995stochastic"/> varying [[integral]] notation in general applies to all integrals here and elsewhere.}}