Notes on Campbell’s theorem
H. Paul Keeler
February 5, 2016
This work is licensed under a “CC BY-SA 3.0” license.
In probability 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 with the intensity function of the point process. One version [9]
of the theorem specifically relates to the Poisson point process and gives a method of
calculating moments as well as Laplace functionals of the process.
A more general result also by the name of Campell’s theorem [12], but also known
as Campbell’s formula [3], entails an integral equation for the aforementioned sum
over a general point process, and not necessarily a Poisson point process. [3].
All the results are employed in probability and statistics with a particular importance in the fields of point processes [7], stochastic geometry [12] and continuum
percolation theory [10], spatial statistics [11, 3].
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Campbell’s theorem and formula
For a point process Φ defined on Euclidean space Rn1 , Campbell’s theorem offers a
way to calculate expectations of a function (with range in the real line Rs) summed
over Φ, namely:
X
f (x).
x∈Φ
The theorem’s name stems from the work [6, 5] of Norman R. Campbell on shot
noise noise in vacuum tubes, which is now considered pioneering in the field of point
processes [7]. More specific instances of this result are also referred to as Campbell’s
theorem in the context of the Poisson point processes [9]. In particular, an equation
stemming from this result, sometimes known as Campbell’s formula [3], holds for
both Poisson and more general (not necessarily simple) point processes [3].
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It can be defined on a more general mathematical space, but often this space is of importance for
models [7].
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1.1
Campbell’s theorem
Another result known as Campbell’s theorem [9] says that for a Poisson point process
Φ and a measurable function f : Rd → R, the sum
X
Σ=
f (x)
x∈Φ
is absolutely convergent with probability one if and only if the integral
Z
min(|f (x)|, 1)Λ(dx) < ∞.
Rd
Provided that this integral is finite, then the theorem asserts that for any complex
value θ the equation
R
f (x)
E(eθΣ ) = e Rd [e −1]Λ(dx) ,
holds if the integral on the right-hand side converges, which is the case for purely
imaginary θ. Moreover
Z
f (x)Λ(dx),
Σ=
Rd
and if this integral converges, then
Z
f (x)2 Λ(dx),
Var(Σ) =
Rd
where Var(Σ) denotes the variance of Σ.
Some results for the Poisson point process follow directly from this theorem including the equation for its Laplace2 functional [9].
1.2
Campbell’s formula
For a general (not necessarily simple) point process Φ with intensity measure Λ(B) =
E[Φ(B)], a result known as as Campbell’s formula [3] or Campbell’s theorem [12, 8, 4],
which gives a way of calculating expectations of sums of measurable functions f with
ranges on the real line. More specifically, for a point process Φ and a measurable
function f : Rd → R, the sum of f over the point process is given by the equation:
"
# Z
X
E
f (x) =
f (x)Λ(dx),
Rd
x∈Φ
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Kingman [9] calls it a ”characteristic functional” but Daley and Vere-Jones [7] and others call it a
”Laplace functional” [12, 1], reserving the term ”characteristic functional” for when θ is imaginary.
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where if one side of the equation is finite, then so is other side [2]. This equation
is essentially an application of Fubini’s theorem [12] and coincides with the aforementioned Poisson case, but holds for a much wide class of point processes, be them
simple or not [3]. Depending on the integral notation3 , this integral may also be written as [2]:
"
# Z
X
E
f (x) =
f (x)Λ(dx),
Rd
x∈Φ
If the intensity measure Λ of a point proces Φ has a density λ, then Campbell’s
formula becomes:
"
# Z
X
E
f (x) =
f (x)λ(x)dx
Rd
x∈Φ
Furthermore, for a stationary point process Φ with constant intensity λ > 0, this
reduces to a volume integral:
"
#
Z
X
E
f (x) = λ
f (x)dx
Rd
x∈Φ
Consequently, these two equations respectively hold for the homogeneous and inhomogeneous Poisson point processes [12].
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Applications
2.1
Laplace functional
For a Poisson point process Φ with intensity measure Λ, the Laplace functional is a
consequence of Campbell’s theorem [9] and is given by [1]:
LΦ = e −
R
Rd
(1−ef (x) )Λ(dx)
,
which for the homogeneous case is:
LΦ = e−λ
R
Rd
(1−ef (x) )dx
.
References
[1] F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks, Volume I - Theory, volume 3, No 3-4 of Foundations and Trends in Networking. NoW
Publishers, 2009.
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As discussed in Chapter 1 of Stoyan, Kendall and Mecke [12], which applies to all other integrals
presented here due to varying integral notation.
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[2] 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.
[3] A. Baddeley, I. Bárány, and R. Schneider. Spatial point processes and their applications. Stochastic Geometry: Lectures given at the CIME Summer School held in
Martina Franca, Italy, September 13-18, 2004, pages 1–75, 2007.
[4] P. Brémaud. Fourier Analysis of Stochastic Processes. Springer, 2014.
[5] N. Campbell. Discontinuities in light emission. In Proc. Cambridge Phil. Soc,
volume 15, page 3, 1909.
[6] N. Campbell. The study of discontinuous phenomena. In Proc. Camb. Phil. Soc,
volume 15, page 310, 1909.
[7] D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes.
Vol. I. Probability and its Applications (New York). Springer, New York, second
edition, 2003.
[8] D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol.
II. Probability and its Applications (New York). Springer, New York, second
edition, 2008.
[9] J. F. C. Kingman. Poisson processes, volume 3. Oxford university press, 1992.
[10] R. Meester and R. Roy. Continuum percolation, volume 119 of cambridge tracts
in mathematics, 1996.
[11] J. Møller and R. P. Waagepetersen. Statistical inference and simulation for spatial
point processes. CRC Press, 2003.
[12] D. Stoyan, W. S. Kendall, and J. Mecke. Stochastic geometry and its applications,
volume 2. Wiley Chichester, 1995.
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