Point process notation: Difference between revisions

Revision as of 13:57, 9 April 2015

Revision as of 06:48, 10 April 2015

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On the other hand, the symbol:

<math> \Phi </math>

represents the number of points of <math> {N}</math> in <math> B</math>. In the context of random measures, one can write:

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:<math> \int_{\textbf{N}} f(x) {N}(dx) </math>

where <math> \textbf{N}</math> is the [[Sample_spaceSample space|space]] of all possible counting measures, hence putting an emphasis on the interpretation of <math> {N}</math> as a random counting measure. An alternative integration notation may be used to write this integral as:

:<math> \int_{\textbf{N}} f \, d{N} </math>

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:<math> E[{N}(B)]=E\left( \sum_{x\in {N}}1_B(x)\right) \qquad \text{or} \qquad \int_{\textbf{N}}\sum_{x\in {N}}1_B(x) P(d{N}). </math>

which is also known as the first [[moment measure]] of <math> {N}</math>. The expectation of such a random sum, known as a ''shot noise process'' in the theory of point processes, can be calculated with [[Campbell's theorem (probability)#Campbell's theorem: general point process|Campbell's theorem]].<ref name="daleyPPI2003"/>.

==Uses in other fields==