The Laplace functional

When working with random variables, some useful tools are functions such as the characteristic function and the moment-generating function, which for a random variable \(Y\) are defined respectively as
$$
\phi_Y(t)= \mathbb{E}\left [ e^{itY} \right ]\,
$$
and
$$
M_Y(t)= \mathbb{E}\left [ e^{tY} \right ]\,,
$$
where the imaginary number \(i=\sqrt{-1}\) and the real variable \(t\in \mathbb{R}\). For continuous random variables, these two respective functions are essentially the Fourier and Laplace transforms of the probability densities. (The moment-generating function  \(M(t)\) may not exist due to the integral not converging to a finite value, whereas the characteristic function \(\phi_Y(t)\) always exists.)

If \(Y\) is a discrete random variable, the probability-generating function is useful, which is defined as
$$
G_Y(z)= \mathbb{E}\left [z^Y \right ]\,.
$$
This function is the Z-transform of the probability mass function of the random variable \(Y\).

Using these tools, results such as sums of random variables and convergence theorems can be proven. There exist equivalent tools which prove useful for studying point processes (and, more generally random measures).

Laplace functional

For a point process \(\Phi \) defined on some underlying space \(\mathbb{S}\), such as \(\mathbb{R}^d\), the Laplace functional is defined as
$$
L_{\Phi}(f)=\mathbb{E}[e^{-\int_{ \mathbb{S}} f(x){\Phi}(dx)}]\,,
$$
where \(f\) is any (Borel) measurable non-negative function on the space \(\mathbb{S}\).

A simple point process is one for which no two or more points coincidence with probability zero. For a simple point process, we can write the random integral (or sum) using set theory notation, giving
$$
\int_{\mathbb{S}} f(x){\Phi}(dx)=\sum\limits_{x\in \Phi} f(x) \,.
$$

Name

Why’s it called a Laplace functional? From its definition, it’s clear that the first half of the name stems from the Laplace transform. Mapping from the space \(\mathbb{S}\), it’s called a functional because it is a function of a non-negative function  \(f\).

Characterization

The Laplace functional characterizes the point process, meaning each point process (or, more generally, random measure) has its own unique Laplace functional. For a given point process, the challenge is to derive the mathematical expression for the Laplace functional by using its definition.

Poisson example

For deriving the Laplace functional, perhaps not surprisingly, one of the easiest one of the easiest point processes is the Poisson point process due to its independence property. For a Poisson process \(\Phi\) with intensity  measure  \(\Lambda\) defined on the state space \(\mathbb{S}\), the Laplace functional is given by
$$
L_{\Phi}(f)=e^{-\int_{ \mathbb{S}} [1-e^{-f(x)}]\,\Lambda(dx) } \,.
$$

If the Poisson point process is homogeneous, then

$$
L_{\Phi}(f)=e^{-\lambda\int_{ \mathbb{S}} [1-e^{-f(x)}]\,dx } \,,
$$

where \(\lambda\) is the intensity function (that is, the average density of points).

Applications

Proof techniques

Given a Laplace functional characterizes a point process, it can be used prove results on the distributions of point processes, where the proofs often simpler. For example, it can used to see what happens when you perform a point process operation on a point process, such as proving that the independent thinning a Poisson point process gives another Poisson point process.  Laplace functionals are used to prove results on the superposition and (random or deterministic) mapping of point processes.

Interference in wireless network models

In the previous post, I covered the concept of the signal-to-interference ratio or SIR in wireless networks. (If noise is included, then then signal-to-interference-plus-noise ratio or just SINR.) Under such wireless network models, the interference term is a type of shot noise of the point process used for the transmitter locations.

Researchers commonly assume Rayleigh fading of the signal energy, which corresponds to the power values randomly varying according to an exponential distribution (due to a square root being taken).  The tail distribution of an exponential variable \(F\) with mean \(\mu\)  is simply \(\mathbb{P}(F>t)= e^{-t/\mu}\).  This means that the exponential assumption and some conditioning arguments lead to Laplace transforms of random variables, including the interference, which can be recast as the Laplace functional of the point process used for the transmitter locations.

Related functionals

For random variables, the characteristic, moment-generating, and probability-generating functions are similarly defined and closely related. We now define two other functionals used for studying point processes.

Characteristic functional

For a point process \(\Phi \) defined on \(\mathbb{S}\), the characteristic functional is defined as
$$
L_{\Phi}(f)=\mathbb{E}[e^{i\int_{ \mathbb{S}} g(x){\Phi}(dx)}]\,,
$$
where \(i=\sqrt{-1}\) and \(g\) is any (Borel) measurable function on the space \(\mathbb{S}\).

Probability-generating functional

For a point process \(\Phi \) defined on \(\mathbb{S}\), the probability-generating functional is defined as
$$
G_{\Phi}(v)=\mathbb{E}[ \prod_{x\in \Phi  } v(x)]\,,
$$
where \(v\) is any bounded non-negative (Borel) measurable function on the space \(\mathbb{S}\) such that \(0\leq v(x)\leq 1\) for any point \(x\in \mathbb{S}\). (Some authors use an alternative definition with a function \(u(x)=1-v(x)\).)

Further reading

There is a Wikipedia article on the Laplace functional.

The usual sources on point processes (and, more generally, random measures) cover Laplace functionals. For example, see section 7.2.1 of the text Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke. The Laplace and other functionals are covered in Section 9.4  of the second volume of An Introduction to the Theory of Point Processes by Daley and Vere-Jones.

Baccelli and Błaszczyszyn use the Laplace to prove some results on the Poisson point process in  Section 1.2 in the first volume of Stochastic Geometry and Wireless Networks.  In an approachable manner, Haenggi details the Laplace and probability-generating fucntionals in Stochastic Geometry for Wireless networks.

The recent book Random Measures, Theory and Applications by Kallenberg also uses Laplace functionals; see Lemma 3.1. Finally, Baccelli, Błaszczyszyn, and Karray use the Laplace functional in the recent book (manuscript) Random Measures, Point Processes, and Stochastic Geometry, but they call it a Laplace transform; see Section 1.3.2, 2.1.1 and 2.2.2, among others.

 

Shot noise

Given a mathematical model based on a point process, a quantity of possible interest is the sum of some function applied to each point of the point process. This random sum is called shot noise, where the name comes from developing mathematical models of the noise measured in old electronic devices, which was likened to shot (used in guns) hitting a surface.

Researchers have long studied shot noise induced by a point process. One particularly application is wireless network models, in which the interference term is an example of shot noise. It is also possible to construct new point processes, called shot noise Cox point processes, by using based on the shot noise of some initial point process.

For such applications, we need a more formal definition of shot noise.

Definition

Shot noise of a point process

We consider a point processes \(\Phi=\{X_i\}_i\) defined on some space \(\mathbb{S}\), which is often \(\mathbb{R}^n\), and a non-negative function \(f\) with the domain \(\mathbb{S}\), so \(f:\mathbb{S} \rightarrow [0,\infty)\). This function is called the response function.

Then the shot noise is defined as
$$
I= \sum_{X_i\in \Phi} f(X_i)\,.
$$

Shot noise of a marked point process

The previous definition of shot noise can be generalized by considering a marked point process \(\Phi’=\{(X_i, M_i)\}_i\), where each point \(X_i\) now has a random mark \(M_i\), which can be a random variable some other random object taking values in some space \(\mathbb{M}\). Then for a response function \(g:\mathbb{S}\times \mathbb{M} \rightarrow [0,\infty)\) , the shot noise is defined as
$$
I’= \sum_{(X_i, M_i)\in \Phi’} g(X_i,M_i)\,.
$$

Properties

Given a point process on a space, like the plane, at any point the shot noise is simply a random variable. If we consider a subset of the space, then shot noise forms a random field, where we recall that a random field is simply a collection of random variables indexed by some set. (By convention, the set tends to be Euclidean space or a manifold). The shot noise can also be considered as a random measure, for example
$$
I(B)= \sum_{X_i\in \Phi\cap B} f(X_i)\,,
$$
where \(B\subseteq \mathbb{S}\). This makes sense as the point process \(\Phi\) is an example of a random (counting) measure.

For Poisson point processes, researchers have studied resulting shot noise random variable or field. For example, given a homogeneous Poisson point process on \(\mathbb{R}^d\), if the response function is a simple power-law \(f(x)=|x|^{\beta}\), where \(\beta> d\), then the resulting shot noise is alpha stable random variable with parameter \(\alpha=d/\beta\).

For a general point process \(\Phi\) with intensity measure \(\Lambda\), the first moment of the shot noise is simply
$$
\mathbb{E}(I)= \int_{\mathbb{S}} f(x) \Lambda (dx) \,.
$$

This is a result of Campbell’s theorem or formula. A similar expression exists for the shot noise of a marked point process.

Some history

Shot noise has been studied for over a century in science. In physics, Walter Schottky did research on shot noise in Germany at the beginning of the 20th century. In the same era, Norman R. Campbell studied shot noise in Britain and wrote two key papers, where one of them contains a result now called Campbell’s theorem or Campbell’s formula, among other names, which is a fundamental result in point process theory. Campbell was a physicist, but his work contains this mathematical result for which he credited the famed pure mathematician G. H. Hardy.

(It’s interesting to note that Hardy claimed years later that, given he did pure mathematics, none of his work would lead to applications, but that claim is simply not true for this and other reasons.)

The work on the physical process of shot noise motivated more probability-oriented papers on shot noise, including:

  • 1944, S. O. Rice, Mathematical Analysis of Random Noise;
  • 1960, Gilbert and Pollak. Amplitude distribution of shot noise;
  • 1971, Daley, The definition of a multi-dimensional generalization of shot noise;
  • 1977, J. Rice, On generalized shot noise;
  • 1990 Lowen and Teich, Power-law shot noise.

Further reading

As a model for interference in wireless networks, shot noise is covered in books such as the two-volume textbooks Stochastic Geometry and Wireless Networks by François Baccelli and Bartek Błaszczyszyn, where the first volume is on theory and the second volume is on applications. Martin Haenggi wrote a very readable introductory book called Stochastic Geometry for Wireless networks.