Campbell's theorem (probability)
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.
Another result by the name of Campell's theorem, but also known as Campbell's formula,:28 entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process. There also exist equations involving moment measures and factorial moment measures 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 processes, stochastic geometry and continuum percolation theory, spatial statistics.
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.
For a point process defined on (d-dimensional) Euclidean space [a], Campbell's theorem offers a way to calculate expectations of a function (with range in the real line R) defined also on and summed over , namely:
where denotes the expectation and set notation is used such that is considered as a random set (see Point process notation). For a point process , Campbell's theorem relates the above expectation with the intensity measure Λ. In relation to a Borel set B the intensity measure of is defined as:
Campbell's theorem: Poisson point process
One version of Campbell's theorem says that for a Poisson point process and a measurable function , the random sum
Provided that this integral is finite, then the theorem further asserts that for any complex value the equation
and if this integral converges, then
where denotes the variance of the random sum .
Campbell's theorem: general point process
A related result for a general (not necessarily simple) point process with intensity measure:
is known as Campbell's formula or Campbell's theorem, which gives a method for calculating expectations of sums of measurable functions with ranges on the real line. More specifically, for a point process and a measurable function , the sum of over the point process is given by the equation:
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 notation[c], this integral may also be written as:
If the intensity measure of a point process has a density , then Campbell's formula becomes:
Stationary point process
For a stationary point process with constant density , Campbell's theorem or formula reduces to a volume integral:
Laplace functional of the Poisson point process
which for the homogeneous case is:
- It can be defined on a more general mathematical space than Euclidean space, but often this space is used for models.
- Kingman calls it a "characteristic functional" but Daley and Vere-Jones and others call it a "Laplace functional", reserving the term "characteristic functional" for when is imaginary.
- As discussed in Chapter 1 of Stoyan, Kendall and Mecke, which applies to all other integrals presented here and elsewhere due to varying integral notation.
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