Campbell’s theorem (formula)

In a previous post, I wrote about the concept of shot noise of a point process. In the simplest terms, shot noise is just the sum of some function over all the points of a point process. The name stems from the original mathematical models of the noise in old electronic devices, which was compared to shot (used in guns) hitting a surface.

In this post I will present a result known as Campbell’s theorem or  Campbell’s formula, which gives the expectation of shot noise as as simple integral expression. This both a general holding for all point processes. It is also useful, as shot noise naturally arises in mathematical models. One application is wireless network models, where the interference term is shot noise.

But to present the main result, I first need to give some basics of point processes, most of which I already covered in this post.

Point process basics

We consider a point processes \(\Phi\) defined on some underlying mathematical space \(\mathbb{S}\), which is often \(\mathbb{R}^n\).  Researchers typically interpret a point process as a random counting measure, resulting in the use of integral and measure theory notation. For example, \(\Phi(B)\) denotes the number of points located in some (Borel measurable) set \(B\), which is a subset of \(\mathbb{S}\).

For point processes, researchers often use a dual notation such that \(\Phi\) denotes both a random set or a random measure.  Then we can write, for example, \(\Phi=\{X_i\}_i\) to stress that \(\Phi\) is a random sets of points.  (Strictly speaking, you can only use the set notation if the point process is simple, meaning that no two points coincide with probability one.)

The first moment measure of a point process, also called the mean measure or intensity measure, is defined as

$$\Lambda(B)= \mathbb{E} [\Phi(B)]. $$

In other words, the first moment measure can be interpreted as the expected number of points of \(\Phi\) falling inside the set \(B \subseteq \mathbb{S}\).

Shot noise definition

We assume a point process \(\Phi=\{X_i\}_i\) is defined on some space \(\mathbb{S}\). We consider a non-negative function \(f\) with the domain \(\mathbb{S}\), so \(f:\mathbb{S} \rightarrow [0,\infty)\).  If the point process \(\Phi\) is a simple, we can use set notation and define the shot noise as

$$
S= \sum_{X_i\in \Phi} f(X_i)\,.
$$

More generally, the shot noise is defined as

$$
S= \int_{ \mathbb{S}} f(x) \Phi(dx)\,.
$$

(We recall that an integral is simply a more general type of sum, which is why the integral sign comes from the letter S.)

Campbell’s theorem

We now state Campbell’s theorem.

Campbell’s theorem says that for any point process \(\Phi\) defined on a space \(\mathbb{S}\) the following formula holds

$$
\mathbb{E}[ S] = \int_{ \mathbb{S}} f(x) \Lambda(dx)\,,
$$

where \(\Lambda= \mathbb{E} [\Phi(B)]\) is the intensity measure of the point process \(\Phi\).

Interpretation

The integral formula is just an application of Fubini’s theorem, as we have simply changed the order of integration.  The formula holds for general processes because it is simply a result on first moments, so it is leveraging the linearity of sums and integrals, including the expectation operator. Put more simply, the sum of parts does equal the whole.

Some history

At the beginning of the 20th century, Norman R. Campbell studied shot noise in Britain and wrote two key papers. In one of these papers appears a version of the result we now called Campbell’s theorem or Campbell’s formula. Interestingly, Campbell was a physicist who credited his mathematical result to renown pure mathematician G. H. Hardy.  Hardy claimed years later that, given he did pure mathematics, none of his work would lead to applications. But Hardy’s claim is simply not true due to this result, as well as his results in number theory.

For some basics on point processes, I suggest the classic text Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke, which covers point processes and the varying notation in Chapters 2 and 4. Haenggi also wrote a very readable introductory book called Stochastic Geometry for Wireless networks, where he gives the basics of point process theory.

Author: Paul Keeler

I am a researcher with interests in mathematical models involving randomness, particularly models with some element of geometry. Much of my work studies wireless networks with a focus on using tools from probability theory such as point processes. I come from Australia, where I call Melbourne home, but I have lived several years in Europe. I grew up in country Queensland and New South Wales.

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