Point process notation: Difference between revisions

Revision as of 19:33, 29 August 2015

Revision as of 20:29, 3 September 2015

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<math> \Phi(B)=n</math>

to denote that there is the set <math> B</math> that contains <math> n</math> points of <math> {N}</math>. In other words, a point process 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">{{Cite journal | doi = 10.1007/1-84628-150-4 | title = Theory of Random Sets | first = Ilya | last = Molchanov| series = Probability and Its Applications | year = 2005 | isbn = 1-85233-892-X | pmid = | pmc = }}</ref>{{rp|106}} and the techniques of random measure theory offering another way to study point processes,<ref name="stoyan1995stochastic"/><ref name="grandell1977point">{{cite jstorjournal | last1 = Grandell | first1 = Jan | year = 1977 | title = Point Processes and Random Measures | journal = Advances in Applied Probability | volume = 9 | issue = 3 | pages = 502-526 | publisher = Applied Probability Trust | jstor = 1426111 | url = | format = | accessdate = }}</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.}}

==Dual notation==