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Point Processes

Basics of Probability

Let $(E, \mathcal E)$ be a measurable space, and let $(E^N, \mathcal E^{\otimes N})$ be the product space equipped with the product $\sigma$-algebra. We will view $\mathbf x = (x_1, \ldots, x_N)$ as the position of $N$ points, or particles in $E$. If the position of the particles is random, then we think of $X_1, X_2, \ldots, X_N$ as $E$-valued random variables representing the random positions of the particles.

In the simplest situation, the particles are indistinguishable, and in this situation there are some events in $\mathcal E^{\otimes N}$ which are not consistent with this. For instance, if $A \in \mathcal E$, then $A \times E^{N-1}$ is in $\mathcal E^{\otimes N}$ represents the event that the “first” particle (the one at $x_1$) is in $A$. But if the particles are indistinguishable, we cannot tell if the first particle is in $A$, but rather the closest we could get would be to tell if one of the particles is in $A$.

In order to capture this indistinguishability we need to restrict our attention to a smaller $\sigma$-algebra, one with events that we could actually observe. The simplest way to do this is to, for $A \in \mathcal E$, define the function $N_A : E^N \rightarrow \mathbb N$ specified by $N_A(\mathbf x) = \#\{ x_1, \ldots, x_N\} \cap A$. That is $N_A$ counts the number of particles in $A$. We may then define the cylinder $\sigma$-algebra on $E^N$ to be $\mathcal C = \sigma\{ N_A : A \in \mathcal E\}$.

Definition: A point process is a probability space $(E^N, \mathcal C, \mathbb P)$ where $\mathcal C$ is the cylinder $\sigma$-algebra on $E^N$.

Given a finite number of disjoint sets $A_1, \ldots, A_M$ in $\mathcal E$ and non-negative integers $n_1, \ldots, n_M$ we call $$\{ N_{A_1} = n_1, \ldots, N_{A_M} = n_M \}$$ a simple cylinder set. The collection of simple cylinder sets forms a $\pi$-system, and to specify $\mathbb P$ on $\mathcal C$ it suffices to specify it on simple cylinder sets.

Joint Intensities & Correlation Functions

One way to get our hands on probabilities of basic cylinder sets are the expectations $\mathbb E[N_{A_1} \cdots N_{A_M}]$ for $A_1, \ldots, A_M$ disjoint sets in $\mathcal E$.

Definition: If there is a measure $\rho_M$ on $(E^M, \mathcal E^{\otimes M})$ so that for all $A_1, \ldots, A_M$ disjoint in $\mathcal E$, $$\mathbb E[N_{A_1} \cdots N_{A_M}] =\rho_M(A_1 \times \cdots \times A_M),$$ then we call $\rho_M$ the $M$th joint intensity measure of the process.

In the case where $E$ is a locally compact abelian group with (some) Haar measure $\lambda$, then there may be a Borel measurable function $R_M : E^M \rightarrow [0,\infty)$ so that for all $A_1, \ldots, A_M$ disjoint Borel sets, $$\mathbb E[N_{A_1} \cdots N_{A_M}] = \int_{A_1 \times \cdots \times A_M} R_M(\mathbf x) \, d\lambda^M(\mathbf x)$$ then we call $R_M$ the $M$th joint intensity function, or the $M$th correlation function, of the process.

When $(E, \mathcal E) = (\mathbb R, \mathcal B(\mathbb R))$ this allows us to compute expectations like $$\mathbb E[N_{A_1} \cdots N_{A_M}] = \int_{A_1} \cdots \int_{A_M} R_M(\mathbf x) \, dx_1 \cdots dx_M$$ via (multiple) Riemann integrals.

The 1-Point Function and the Spatial Density of Particles

From the definition of intensity measures, the first intensity satisfies $$N = \mathbb E[N_E] = \int d\rho_1,$$ and if $E$ has reference measure and if the first intensity functions is defined, then $$\frac{1}{N} \int R_1(x) \, dx = 1.$$ That is $\rho_1/N$ is a probability measure on $(E, \mathcal E)$ and $R_1/N$ is the density function for $\rho_1/N$. This gives the following theorem,

Theorem: Let $A \in \mathcal E$, then the probability that at least one particle lies in $A$ is given by $$\mathbb P\{ \{X_1, \ldots, X_N\} \cap A \neq \emptyset \} = \frac{1}{N} \int_A d\rho_1 = \frac1N \int_A R_1(x) \, dx,$$ where the last expression is valid only if the first intensity function is defined.

$R_1/N$ is sometimes called the 1-point function, and it gives the spatial density of particles in $E$.

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