The Laplace functional

When working with random variables, a couple useful tools are 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, a useful object is the probability-generating function, 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 \(Y\).

By 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 two or more points coincide in location 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 functionals 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.