en>K9re11: removed Category:Mathematical theorems; added Category:Theorems in combinatorics using HotCat

2014-12-11T18:41:43Z

removed Category:Mathematical theorems; added Category:Theorems in combinatorics using HotCat

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en>P199: no discussion, no merge

2014-02-12T15:58:15Z

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en>Michael Hardy: proper initial context setting; cleaned up some of the notatino per WP:MOSMATH

2012-02-25T19:41:50Z

proper initial context setting; cleaned up some of the notatino per WP:MOSMATH

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{{about|random point processes||Campbell's theorem (geometry)}}

In [[probability theory]] and [[statistics]], '''Campell's theorem''' can refer to a particular [[equation]] or set of results relating to the [[Expected value|expectation]] of a [[Function (mathematics)|function]] summed over a [[point process]] to an [[integral]] involving the [[Moment_measure#First_moment_measure|intensity measure]] of the point process, which allows for the calculation of [[expected value]] and [[variance]] of the [[random]] [[sum]]. One version<ref name="kingman1992poisson">{{cite book|title=Poisson Processes|page=28|first=John|last=Kingman|authorlink=John Kingman|publisher=Oxford Science Publications|year=1993|isbn=0-19-853693-3}}</ref> of the theorem specifically relates to the [[Poisson point process]] and gives a method for calculating moments as well as [[Laplace functional]]s of the process.

Another result by the name of Campell's theorem,<ref name="stoyan1995stochastic">D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.</ref> but also known as '''Campbell's formula''',<ref name="baddeley2007spatial">{{cite doi|10.1007/978-3-540-38175-4_1}}</ref>{{rp|28}} entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process.<ref name="baddeley2007spatial"/> There also exist equations involving [[moment measure]]s and [[factorial moment measure]]s that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the related fields of [[point process]]es,<ref name="daleyPPI2003">{{cite doi|10.1007/b97277}}</ref> [[stochastic geometry]]<ref name="stoyan1995stochastic"/> and [[continuum percolation theory]],<ref name="meester1996continuum">R. Meester and R. Roy. Continuum percolation, volume 119 of Cambridge tracts in mathematics, 1996.

</ref> [[spatial statistics]].<ref name="baddeley2007spatial"/><ref name="moller2003statistical">{{cite doi|10.1201/9780203496930}}</ref>

The theorem's name stems from the work<ref name="campbell1909study">{{Cite journal|last=Campbell|first= N.| authorlink=Norman Robert Campbell |year=1909|journal= Proc. Cambr. Phil. Soc. |volume=15 |title=The study of discontinuous phenomena|pages= 117–136|url=http://www.archive.org/details/proceedingsofcam15190810camb}}</ref><ref name="campbell1909discontinuities">{{cite journal|last=Campbell|first= N.| authorlink = Norman Robert Campbell |year=1910|journal= Proc. Cambr. Phil. Soc. |volume=15 |title=Discontinuities in light emission
|pages=310–328|url=http://www.archive.org/details/proceedingsofcam15190810camb}}</ref> by [[Norman Robert Campbell|Norman R. Campbell]] on [[shot noise]],<ref name="daleyPPI2003"/> which was partly inspired by the work of [[Ernest Rutherford]] and [[Hans Geiger]] on [[alpha particle]] detection, where the [[Poisson point process]] arose as a solution to a family of differential equations by [[Harry Bateman]].<ref name="stirzaker2000advice">{{cite jstor|3621649}}</ref> In Campbell's work, he presents the [[Moment (mathematics)|moments]] and [[Moment-generating function|generating functions]] of the random sum of a Poisson process on the real line, but remarks that the main mathematical argument was due to [[G. H. Hardy]], which has inspired the result to be sometimes called the '''Campbell-Hardy theorem'''.<ref name="stirzaker2000advice"/><ref>{{cite book|last=Grimmett G. and Stirzaker D.|title=Probability and random processes|year=2001|publisher=Oxford University Press|pages=290}}</ref>

==Background==

For a point process <math>{N}</math> defined on (''d''-dimensional) [[Euclidean space]] <math>\textbf{R}^d </math> {{efn|It can be defined on a more general mathematical space than Euclidean space, but often this space is used for models.<ref name="daleyPPI2003"/>}}, Campbell's theorem offers a way to calculate expectations of a function <math>f</math> (with [[range (mathematics)|range]] in the real line '''R''') defined also on <math>\textbf{R}^d </math> and summed over <math>{N}</math>, namely:

:<math>E[ \sum_{x\in {N}}f(x)] </math>,

where <math>E </math> denotes the expectation and set notation is used such that <math>{N}</math> is considered as a random set (see [[Point process notation]]). For a point process <math>{N}</math>, Campbell's theorem relates the above expectation with the intensity measure ''Λ''. In relation to a [[Borel set]] ''B'' the intensity measure of <math>{N}</math> is defined as:

:<math>\Lambda(B)=E[ {N}(B) ] </math>,

where the [[Measure (mathematics)|measure]] notation is used such that <math>{N}</math> is considered a random [[counting measure]]. The quantity ''Λ(B)'' can be interpreted as the average number of points of <math>{N}</math> located in the set ''B''.

==Campbell's theorem: Poisson point process==

One version of Campbell's theorem<ref name="kingman1992poisson"/> says that for a Poisson point process <math>{N}</math> and a measurable function <math> f: \textbf{R}^d\rightarrow \textbf{R}</math>, the random sum

:<math> \Sigma=\sum_{x\in {N}}f(x) </math>

is [[absolutely convergent]] with [[Almost surely|probability one]] [[if and only if]] the integral

:<math> \int_{ \textbf{R}^d} \min(|f(x)|,1)\Lambda (dx) < \infty. </math>

Provided that this integral is finite, then the theorem further asserts that for any [[Complex Number|complex]] value <math>\theta</math> the equation

:<math> E(e^{\theta\Sigma})=\textrm{exp} \left(\int_{\textbf{R}^d} [e^{f(x)}-1]\Lambda (dx)\right), </math>

holds if the integral on the right-hand side [[convergence (mathematics)|converges]], which is the case for purely [[Imaginary number|imaginary]] <math>\theta</math>. Moreover

:<math> \Sigma=\int_{\textbf{R}^d} f(x)\Lambda (dx), </math>

and if this integral converges, then

:<math> \text{Var}(\Sigma)=\int_{\textbf{R}^d} f(x)^2\Lambda (dx), </math>

where <math> \text{Var}(\Sigma)</math> denotes the variance of the random sum <math> \Sigma</math>.

From this theorem some expectation results for the [[Poisson point process]] follow directly including its Laplace functional.<ref name="kingman1992poisson"/> {{efn|Kingman<ref name="kingman1992poisson"/> calls it a "characteristic functional" but Daley and Vere-Jones<ref name="daleyPPI2003"/> and others call it a "Laplace functional",<ref name="stoyan1995stochastic"/><ref name="BB1">{{cite doi|10.1561/1300000006}}</ref> reserving the term "characteristic functional" for when <math> \theta</math> is imaginary.}}

==Campbell's theorem: general point process==

A related result for a general (not necessarily simple) point process <math>{N}</math> with intensity measure:

:<math> \Lambda (B)= E[{N}(B)] , </math>

is known as '''Campbell's formula'''<ref name="baddeley2007spatial"/> or '''Campbell's theorem''',<ref name="stoyan1995stochastic"/><ref name="daleyPPII2008">{{cite doi|10.1007/978-0-387-49835-5}}</ref> which gives a method for calculating expectations of sums of [[measurable function]]s <math> f</math> with [[Range (mathematics)|ranges]] on the [[real line]]. More specifically, for a point process <math> {N}</math> and a measurable function <math> f: \textbf{R}^d\rightarrow \textbf{R}</math>, the sum of <math> f</math> over the point process is given by the equation:

:<math> E\left[\sum_{x\in {N}}f(x)\right]=\int_{\textbf{R}^d} f(x)\Lambda (dx), </math>

where if one side of the equation is finite, then so is the other side.<ref name="baddeley1999crash">A. Baddeley. A crash course in stochastic geometry. ''Stochastic Geometry: Likelihood and Computation Eds OE Barndorff-Nielsen, WS Kendall, HNN van Lieshout (London: Chapman and Hall) pp'', pages 1--35, 1999.

</ref> This equation is essentially an application of [[Fubini's theorem]]<ref name="stoyan1995stochastic"/> and coincides with the aforementioned Poisson case, but holds for a much wider class of point processes, simple or not.<ref name="baddeley2007spatial"/> Depending on the integral notation{{efn|As discussed in Chapter 1 of Stoyan, Kendall and Mecke,<ref name="stoyan1995stochastic"/> which applies to all other integrals presented here and elsewhere due to varying integral notation.}}, this integral may also be written as:<ref name="baddeley1999crash"/>

:<math> E\left[\sum_{x\in {N}}f(x)\right]=\int_{\textbf{R}^d} fd\Lambda , </math>

If the intensity measure <math> \Lambda</math> of a point process <math> {N}</math> has a density <math> \lambda(x) </math>, then Campbell's formula becomes:

:<math> E\left[\sum_{x\in {N}}f(x)\right]= \int_{\textbf{R}^d} f(x)\lambda(x)dx </math>

===Stationary point process===

For a stationary point process <math>{N}</math> with constant intensity <math> \lambda>0</math>, '''Campbell's theorem''' or '''formula''' reduces to a volume integral:

:<math> E\left[\sum_{x\in {N}}f(x)\right]=\lambda \int_{\textbf{R}^d} f(x)dx </math>

This equation naturally holds for the homogeneous Poisson point processes, which is an example of a [[stationary stochastic process]].<ref name="stoyan1995stochastic"/>

==Applications==

===Laplace functional of the Poisson point process===

For a Poisson point process <math> {N}</math> with intensity measure <math> \Lambda</math>, the [[Laplace functional]] is a consequence of Campbell's theorem<ref name="kingman1992poisson"/> and is given by:<ref name="BB1"/>

:<math> L_{{N}}=e^{-\int_{\textbf{R}^d}(1-e^{ f(x)})\Lambda(dx)}, </math>

which for the homogeneous case is:

:<math> L_{{N}}=e^{-\lambda\int_{\textbf{R}^d}(1-e^{ f(x)})dx}. </math>

==Notes==
{{notelist}}

==References==
<references/>

[[Category:Probability theorems]]

Labelled enumeration theorem - Revision history

en>K9re11: removed Category:Mathematical theorems; added Category:Theorems in combinatorics using HotCat

en>P199: no discussion, no merge

en>Michael Hardy: proper initial context setting; cleaned up some of the notatino per WP:MOSMATH