Campbell's theorem (probability)
Campbell's theorem (probability)
Campbell's theorem, also called Campbell–Hardy theorem, is a key result in probability that connects the expected value of a sum taken over the points of a random point process to an integral involving the process’s mean measure. This makes it easier to compute the expected value and variance of random sums.
General version
- Let N be a point process on Euclidean space R^d with intensity (mean) measure Λ, defined by Λ(B) = E[N(B)] for any Borel set B.
- For any measurable function f: R^d → R, the sum S = sum of f(x) over all points x in N is considered. Campbell's theorem says that if S is well-defined (finite almost surely when the integral is finite), then:
E[S] = ∫_{R^d} f(x) dΛ(x).
If Λ has a density λ(x), then this becomes:
E[S] = ∫_{R^d} f(x) λ(x) dx.
- In particular, if the integral is finite, then the sum S is finite almost surely.
Poisson point process and the Laplace functional
- A common special case is the Poisson point process. For a nonnegative function f, the Laplace functional is defined as:
L_N(sf) = E[exp(-s ∑_{x in N} f(x))], for s ≥ 0.
- Campbell's theorem implies:
L_N(sf) = exp( - ∫_{R^d} (1 - e^{-s f(x)}) dΛ(x) ).
- In the homogeneous Poisson case, where Λ(dx) = λ dx (constant density λ), this simplifies to:
L_N(sf) = exp( -λ ∫_{R^d} (1 - e^{-s f(x)}) dx ).
- Moments of the sum S in the Poisson case follow similarly:
E[S] = ∫_{R^d} f(x) dΛ(x), and Var(S) = ∫_{R^d} f(x)^2 dΛ(x).
If Λ has density λ(x), then E[S] = ∫ f(x) λ(x) dx and Var(S) = ∫ f(x)^2 λ(x) dx.
Background, applications, and generalizations
- N is a random counting measure (a way to model random points in space). Campbell's formula gives a straightforward way to compute the expectation of sums of functions evaluated at those random points.
- Applications are wide, including random sums in shot noise, models of interference in wireless networks, and sums of synaptic inputs in neuroscience. These use Campbell’s theorem to move from random sums to deterministic integrals.
- If you want sums that depend on more than one point (not just a single point’s contribution), you need more advanced versions involving moment measures or factorial moment measures. For sums over the entire point process, Palm calculus or generalized Campbell theorems are used.
Notes and history
- Campbell's theorem is named after Norman R. Campbell, who studied shot noise in thermionic devices. A related result for Poisson processes is sometimes called Campbell’s Poisson theorem or Campbell–Hardy theorem.
- The theorem holds in broad settings and can be written in different notations, but the core idea remains: the expected sum over a point process can be computed by integrating the summand against the process’s intensity measure.
This page was last edited on 28 January 2026, at 16:33 (CET).