References: [Last and Penrose, 2018, Ch. 2, 6.1, 6.2]; [Baccelli, Blaszczyszyn, and Karray, 2024, Ch. 1.1, 1.6]
There are two equivalent ways to define a point process on some space \(E\):
Definition (Locally Finite Sets). A subset \(A \subset E\) is said to be locally finite if \[ \#(A \cap C) < \infty, \quad \forall C \subset E \text{ compact}. \] Here \(\#(\cdot)\) denotes the number of elements in a set.
Definition (2) is more general, but with additional assumptions, we can understand and interpret a point process as (1) or (2). In this lecture, we will define a point process as (2) and later show how (1) and (2) are related.
Let \(E\) be a metric space. Define:
In this class, we set \(E = \mathbb{R}^d\) or \(E=\mathcal{F}(\mathbb{R}^d)\), which is the space of all closed subsets of \(\mathbb{R}^d\).
Definition (Locally Finite Measures). A measure \(\mu\) on \((E, \mathcal{B})\) is locally finite if \[ \mu(C) < \infty, \quad \forall C \in \mathcal{C}. \]
Definition (Counting Measures). A counting measure \(\mu\) on \((E, \mathcal{B})\) is a locally finite measure such that \[ \mu(B) \in \mathbb{N}_0 \cup \{\infty\}, \quad \forall B \in \mathcal{B}. \]
Denote by \(\mathcal{M}(E)\) the set of all counting measures on \(E\).
The \(\sigma\)-algebra on \(\mathcal{M}(E)\) is generated by the sets \[ S_{B,k} = \{ \mu \in \mathcal{M}(E) : \mu(B) = k \}, \quad B \in \mathcal{B}(E), \; k \in \mathbb{N}_0 \cup \{\infty\}. \] This \(\sigma\)-algebra, often denoted by \(\mathscr{M}(E)\), is the smallest \(\sigma\)-algebra on \(\mathcal{M}(E)\) such that all mappings \[ \pi_B: \mathcal{M}(E) \to \mathbb{N}_0 \cup \{\infty\}, \quad \mu \mapsto \mu(B) \] are measurable for every \(B \in \mathcal{B}(E)\).
Example (Dirac Measures). If \(x \in E\), then \[ \mu = \delta_x \] is a counting measure, where \[ \delta_x(B) = \begin{cases} 1, & x \in B \\ 0, & x \notin B \end{cases} \]
Example (Sum of Dirac Measures). Let \(\{x_i\}_{i=1}^n, n \in \NN\) be a locally finite subset of \(E\). Then \[ \mu = \sum_{i=1}^n \delta_{x_i} \] is a counting measure.
Question: Can any counting measure be written as a sum of Dirac measures?
Answer: In general, no (see [last2018lectures,Ch.2]). However, if we assume that \(E\) is a complete and separable metric space (CSMS), then yes.
Definition (Separable Space). A metric space is separable if it contains a countable dense subset.
Definition (Complete Space). A metric space is complete if every Cauchy sequence converges.
Proposition. Let \(E\) be a CSMS. Then there exist measurable maps \(\pi_i: \mathcal{M}(E) \to E, i\in\mathbb{N},\) such that for all \(\mu \in \mathcal{M}(E)\), \[ \mu = \sum_{i} \delta_{\pi_i(\mu)}. \]
Proof. (See [last2018lectures,Prop.6.2 & 6.3])
\(\square\)
Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space and let \(E\) be a CSMS.
Definition (Point Processes). A point process \(\Phi\) on \(E\) is a random element of \(\left(\mathcal{M}(E), \mathscr{M}(E)\right)\), i.e. a measurable mapping \[ \Phi: \Omega \to \mathcal{M}(E). \]
For a point process \(\Phi\) on \(E\) and a Borel set \(B \in \mathcal{B}(E)\), define the mapping \[ \Phi(B): \Omega \to \NN, \quad \omega \mapsto \Phi(\omega)(B). \]
Then \(\Phi(B)\) is a random variable taking values in \(\NN\) for all \(B \in \mathcal{B}(E)\), which implies \[ \{\Phi(B)\}_{B \in \mathcal{B}(E)} \] is a stochastic process with values in \(\NN\) indexed by Borel sets \(\mathcal{B}(E)\).
Example. Let \(X\) be a random element of \(E\). Then \[ \Phi = \delta_X \] is a point process.
Example (Binomial point process). Let \(X_1, \dots, X_n\) be i.i.d. random elements in \(E\) with distribution \(\mathbb{Q}\). Then \[ \Phi = \sum_{i=1}^n \delta_{X_i} \] is a point process called a binomial point process. For any \(B \in \mathcal{B}(E)\), \[ \mathbb{P}(\Phi(B) = k) = \binom{n}{k} (\mathbb{Q}(B))^k (1 - \mathbb{Q}(B))^{n-k}, \quad k = 0, 1, \dots, n. \]
Example (Mixed binomial point process). Let \(N\) be a random variable taking values in \(\mathbb{N}_0\) and independent of an i.i.d. sequence \(\{X_i\}_{i \ge 1}\) with distribution \(\mathbb{Q}\). Then \[ \Phi = \sum_{i=1}^{N} \delta_{X_i} \] is called a mixed binomial point process.
Proposition [ref] gives the following corollary:
Corollary. For every point process \(\Phi\) on \(E\), there exist random variables \(\{X_i\}_{i \ge 1}\) in \(E\) and a random variable \(N \in \mathbb{N}_0 \cup \{\infty\}\) such that \[ \Phi = \sum_{i=1}^{N} \delta_{X_i}. \]
Note: This representation does not guarantee that \(\Phi\) is simple, since the points \(X_i\) are not necessarily distinct. Multiplicities can occur.
Definition (Support of a counting measure). For \(\mu \in \mathcal{M}(E)\), the support of \(\mu\) is a locally finite subset of \(E\) defined as \[ \mathrm{Supp}(\mu) := \{\, x \in E : \mu(\{x\}) \ge 1 \,\}. \]
Definition (Simple counting measure and point process). A counting measure \(\mu\) is called simple if \[ \mu(\{x\}) \le 1 \quad \text{for all } x \in E. \]
A point process \(\Phi\) is simple if \[ \mathbb{P}(\Phi \text{ is simple}) = 1. \]
If a point process \(\Phi\) is simple, we can identify it with its support. We also have
Proposition (Representation of a simple point process). If a point process on a CSMS \(E\) is simple, i.e. \[ \Phi(\{x\}) \le 1 \quad \forall x \in E, \] then we can write \[ \Phi = \sum_{i=1}^{\Phi(E)} \delta_{X_i}, \] where \(\{ X_i \}_{i=1}^{\Phi(E)}\) form a random locally finite subset of \(E\).