Campbell's theorem (probability)

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This article is about random point processes. For other uses, see 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 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]


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[edit]

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[edit]

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[edit]

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]


Laplace functional of the Poisson point process[edit]

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].


  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.


  1. ^ a b c d e Kingman, John (1993). Poisson Processes. Oxford Science Publications. p. 28. ISBN 0-19-853693-3. 
  2. ^ a b c d e f g D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. Stochastic geometry and its applications, volume 2. Wiley Chichester, 1995.
  3. ^ a b c d e 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.