Labelled enumeration theorem

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In probability theory and statistics, Campell's theorem can refer to a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the intensity measure of the point process, which allows for the calculation of expected value and variance of the random sum. One version of the theorem specifically relates to the Poisson point process and gives a method for calculating moments as well as Laplace functionals of the process.

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The theorem's name stems from the work by Norman R. Campbell on shot noise, 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. In Campbell's work, he presents the moments and 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.

Background

For a point process $N$ defined on (d-dimensional) Euclidean space ${\mathbf{\text{R}}}^d$ Template:Efn, Campbell's theorem offers a way to calculate expectations of a function $f$ (with range in the real line R) defined also on ${\mathbf{\text{R}}}^d$ and summed over $N$, namely:

$E[\sum _{x\in N}f(x)]$,

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

$\mathrm{\Lambda }(B)=E[N(B)]$,

where the measure notation is used such that $N$ is considered a random counting measure. The quantity Λ(B) can be interpreted as the average number of points of $N$ located in the set B.

Campbell's theorem: Poisson point process

One version of Campbell's theorem says that for a Poisson point process $N$ and a measurable function $f:{\mathbf{\text{R}}}^d\to \mathbf{\text{R}}$, the random sum

$\mathrm{\Sigma }=\sum _{x\in N}f(x)$

is absolutely convergent with probability one if and only if the integral

$\underset{{\mathbf{\text{R}}}^d}{\int }\min (|f(x)|,1)\mathrm{\Lambda }(dx)<\mathrm{\infty }.$

Provided that this integral is finite, then the theorem further asserts that for any complex value $\theta$ the equation

$E(e^{\theta \mathrm{\Sigma }})=\text{<mrow data-mjx-texclass="ORD">exp</mrow>}(\underset{{\mathbf{\text{R}}}^d}{\int }[e^{f(x)}-1]\mathrm{\Lambda }(dx)),$

holds if the integral on the right-hand side converges, which is the case for purely imaginary $\theta$. Moreover

$\mathrm{\Sigma }=\underset{{\mathbf{\text{R}}}^d}{\int }f(x)\mathrm{\Lambda }(dx),$

and if this integral converges, then

$\text{Var}(\mathrm{\Sigma })=\underset{{\mathbf{\text{R}}}^d}{\int }f(x{)}^2\mathrm{\Lambda }(dx),$

where $\text{Var}(\mathrm{\Sigma })$ denotes the variance of the random sum $\mathrm{\Sigma }$.

From this theorem some expectation results for the Poisson point process follow directly including its Laplace functional. Template:Efn

Campbell's theorem: general point process

A related result for a general (not necessarily simple) point process $N$ with intensity measure:

$\mathrm{\Lambda }(B)=E[N(B)],$

is known as Campbell's formula or Campbell's theorem, which gives a method for calculating expectations of sums of measurable functions $f$ with ranges on the real line. More specifically, for a point process $N$ and a measurable function $f:{\mathbf{\text{R}}}^d\to \mathbf{\text{R}}$, the sum of $f$ over the point process is given by the equation:

$E[\sum _{x\in N}f(x)]=\underset{{\mathbf{\text{R}}}^d}{\int }f(x)\mathrm{\Lambda }(dx),$

where if one side of the equation is finite, then so is the other side. This equation is essentially an application of Fubini's theorem and coincides with the aforementioned Poisson case, but holds for a much wider class of point processes, simple or not. Depending on the integral notationTemplate:Efn, this integral may also be written as:

$E[\sum _{x\in N}f(x)]=\underset{{\mathbf{\text{R}}}^d}{\int }fd\mathrm{\Lambda },$

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

$E[\sum _{x\in N}f(x)]=\underset{{\mathbf{\text{R}}}^d}{\int }f(x)\lambda (x)dx$

Stationary point process

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

$E[\sum _{x\in N}f(x)]=\lambda \underset{{\mathbf{\text{R}}}^d}{\int }f(x)dx$

This equation naturally holds for the homogeneous Poisson point processes, which is an example of a stationary stochastic process.

Applications

Laplace functional of the Poisson point process

For a Poisson point process $N$ with intensity measure $\mathrm{\Lambda }$, the Laplace functional is a consequence of Campbell's theorem and is given by:

$L_N=e^{-\underset{{\mathbf{\text{R}}}^d}{\int }(1-e^{f(x)})\mathrm{\Lambda }(dx)},$

which for the homogeneous case is:

$L_N=e^{-\lambda \underset{{\mathbf{\text{R}}}^d}{\int }(1-e^{f(x)})dx}.$