Campbell's theorem (probability)

For a point process ${\displaystyle N}$ defined on d-dimensional Euclidean space ${\displaystyle {\textbf {R}}^{d}}$,[lower-alpha 1] Campbell's theorem offers a way to calculate expectations of a real-valued function ${\displaystyle f}$ defined also on ${\displaystyle {\textbf {R}}^{d}}$ and summed over ${\displaystyle N}$, namely:

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

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

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

where the measure notation is used such that ${\displaystyle N}$ is considered a random counting measure. The quantity ${\displaystyle \Lambda (B)}$ can be interpreted as the average number of points of the point process ${\displaystyle N}$ located in the set B.

One version of Campbell's theorem is for a general (not necessarily simple) point process ${\displaystyle N}$ with intensity measure:

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

is known as Campbell's formula[2] or Campbell's theorem,[1][12][13] which gives a method for calculating expectations of sums of measurable functions ${\displaystyle f}$ with ranges on the real line. More specifically, for a point process ${\displaystyle N}$ and a measurable function ${\displaystyle f:{\textbf {R}}^{d}\rightarrow {\textbf {R}}}$, the sum of ${\displaystyle f}$ over the point process is given by the equation:

${\displaystyle E\left[\sum _{x\in N}f(x)\right]=\int _{{\textbf {R}}^{d}}f(x)\Lambda (dx),}$

where if one side of the equation is finite, then so is the other side.[14] This equation is essentially an application of Fubini's theorem[1] and it holds for a wide class of point processes, simple or not.[2] Depending on the integral notation,[lower-alpha 2] this integral may also be written as:[14]

${\displaystyle \operatorname {E} \left[\sum _{x\in N}f(x)\right]=\int _{{\textbf {R}}^{d}}f\,d\Lambda ,}$

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

${\displaystyle \operatorname {E} \left[\sum _{x\in N}f(x)\right]=\int _{{\textbf {R}}^{d}}f(x)\lambda (x)\,dx}$

Campbell's theorem for general point processes gives a method for calculating the expectation of a function of a point (of a point process) summed over all the points in the point process. These random sums over point processes have applications in many areas where they are used as mathematical models.

For general point processes, other more general versions of Campbell's theorem exist depending on the nature of the random sum and in particular the function being summed over the point process.

Another version of Campbell's theorem[7] says that for a Poisson point process ${\displaystyle N}$ with intensity measure ${\displaystyle \Lambda }$ and a measurable function ${\displaystyle f:{\textbf {R}}^{d}\rightarrow {\textbf {R}}}$, the random sum

${\displaystyle S=\sum _{x\in N}f(x)}$

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

${\displaystyle \int _{{\textbf {R}}^{d}}\min(|f(x)|,1)\Lambda (dx)<\infty .}$

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

${\displaystyle E(e^{\theta S})=\exp \left(\int _{{\textbf {R}}^{d}}[e^{\theta f(x)}-1]\Lambda (dx)\right),}$

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

${\displaystyle E(S)=\int _{{\textbf {R}}^{d}}f(x)\Lambda (dx),}$

and if this integral converges, then

${\displaystyle \operatorname {Var} (S)=\int _{{\textbf {R}}^{d}}f(x)^{2}\Lambda (dx),}$

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

From this theorem some expectation results for the Poisson point process follow, including its Laplace functional.[7] [lower-alpha 3]