# 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[1] 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,[2] but also known as Campbell's formula,[3]:28 entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process.[3] 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,[4] stochastic geometry[2] and continuum percolation theory,[5] spatial statistics.[3][6]

The theorem's name stems from the work[7][8] by Norman R. Campbell on shot noise,[4] 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.[9] 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.[9][10]

## Background

For a point process ${N}$ defined on (d-dimensional) Euclidean space $\textbf{R}^d$ [a], Campbell's theorem offers a way to calculate expectations of a function $f$ (with range in the real line R) defined also on $\textbf{R}^d$ 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:

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

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[1] says that for a Poisson point process ${N}$ and a measurable function $f: \textbf{R}^d\rightarrow \textbf{R}$, the random sum

$\Sigma=\sum_{x\in {N}}f(x)$

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

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

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

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

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

$\Sigma=\int_{\textbf{R}^d} f(x)\Lambda (dx),$

and if this integral converges, then

$\text{Var}(\Sigma)=\int_{\textbf{R}^d} f(x)^2\Lambda (dx),$

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

From this theorem some expectation results for the Poisson point process follow directly including its Laplace functional.[1] [b]

## Campbell's theorem: general point process

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

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

is known as Campbell's formula[3] or Campbell's theorem,[2][12] 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 $f: \textbf{R}^d\rightarrow \textbf{R}$, the sum of $f$ over the point process is given by the equation:

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

where if one side of the equation is finite, then so is the other side.[13] This equation is essentially an application of Fubini's theorem[2] and coincides with the aforementioned Poisson case, but holds for a much wider class of point processes, simple or not.[3] Depending on the integral notation[c], this integral may also be written as:[13]

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

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

$E\left[\sum_{x\in {N}}f(x)\right]= \int_{\textbf{R}^d} f(x)\lambda(x)dx$

### Stationary point process

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

$E\left[\sum_{x\in {N}}f(x)\right]=\lambda \int_{\textbf{R}^d} f(x)dx$

This equation naturally holds for the homogeneous Poisson point processes, which is an example of a stationary stochastic process.[2]

## Applications

### Laplace functional of the Poisson point process

For a Poisson point process ${N}$ with intensity measure $\Lambda$, the Laplace functional is a consequence of Campbell's theorem[1] and is given by:[11]

$L_{{N}} := E\bigl[ e^{ \sum_{x \in N} f(x) } \bigr] =\exp \Bigl[-\int_{\textbf{R}^d}(1-e^{ f(x)})\Lambda(dx) \Bigr],$

which for the homogeneous case is:

$L_{{N}}=\exp\Bigl[-\lambda\int_{\textbf{R}^d}(1-e^{ f(x)})dx \Bigr].$

## Notes

1. ^ It can be defined on a more general mathematical space than Euclidean space, but often this space is used for models.[4]
2. ^ Kingman[1] calls it a "characteristic functional" but Daley and Vere-Jones[4] and others call it a "Laplace functional",[2][11] reserving the term "characteristic functional" for when $\theta$ is imaginary.
3. ^ As discussed in Chapter 1 of Stoyan, Kendall and Mecke,[2] which applies to all other integrals presented here and elsewhere due to varying integral notation.

## References

1. Kingman, John (1993). Poisson Processes. Oxford Science Publications. p. 28. ISBN 0-19-853693-3.
2. D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. Stochastic geometry and its applications, volume 2. Wiley Chichester, 1995.
3. Baddeley, A.; Barany, I.; Schneider, R.; Weil, W. (2007). "Spatial Point Processes and their Applications". Stochastic Geometry. Lecture Notes in Mathematics 1892. p. 1. doi:10.1007/978-3-540-38175-4_1. ISBN 978-3-540-38174-7. edit
4. ^ a b c d Daley, D. J.; Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Probability and its Applications. doi:10.1007/b97277. ISBN 0-387-95541-0. edit
5. ^ R. Meester and R. Roy. Continuum percolation, volume 119 of Cambridge tracts in mathematics, 1996.
6. ^ Moller, J.; Plenge Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. C&H/CRC Monographs on Statistics & Applied Probability 100. doi:10.1201/9780203496930. ISBN 978-1-58488-265-7. edit
7. ^ Campbell, N. (1909). "The study of discontinuous phenomena". Proc. Cambr. Phil. Soc. 15: 117–136.
8. ^ Campbell, N. (1910). "Discontinuities in light emission". Proc. Cambr. Phil. Soc. 15: 310–328.
9. ^ a b Stirzaker, David (2000). "Advice to Hedgehogs, or, Constants Can Vary". The Mathematical Gazette (197-210) 84 (500): 197–210. JSTOR 3621649. edit
10. ^ Grimmett G. and Stirzaker D. (2001). Probability and random processes. Oxford University Press. p. 290.
11. ^ a b Baccelli, F. O. (2009). "Stochastic Geometry and Wireless Networks: Volume I Theory". Foundations and Trends® in Networking 3 (3–4): 249–449. doi:10.1561/1300000006. edit
12. ^ Daley, D. J.; Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Probability and Its Applications. doi:10.1007/978-0-387-49835-5. ISBN 978-0-387-21337-8. edit
13. ^ a b 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.