Let \(E = \mathbb{R}^d\).
Definition (Product density). Suppose that the factorial moment measure \(\alpha^{(n)}\) is locally finite and absolutely continuous on \((\mathbb{R}^d)^n\).
Then \(\alpha^{(n)}\) admits a density \(\rho^{(n)}\), called the \(n\)-th order product density (also called joint density).
That is, by Campbell’s theorem, \[ \mathbb{E}\left[ \sum_{(x_1, \dots, x_n) \in \Phi^{(n)}} f(x_1, \dots, x_n) \right] = \int_{(\mathbb{R}^d)^n} f(x_1, \dots, x_n) \rho^{(n)}(x_1, \dots, x_n) \, dx_1 \dots dx_n. \]
The quantity \[ \rho^{(n)}(x_1, \dots, x_n) \, dx_1 \dots dx_n \] can be interpreted as the probability that there is a point of \(\Phi\) in each infinitesimal region \(x_i + dx_i\).
Example. For measurable sets \(B_1, \dots, B_n \subset \mathbb{R}^d\), we have \[ \mathbb{E}\left[ \Phi^{(n)}(B_1 \times \dots \times B_n) \right] = \int_{(\mathbb{R}^d)^n} \mathbf{1}\{ x_1 \in B_1, \dots, x_n \in B_n \} \rho^{(n)}(x_1, \dots, x_n) \, dx_1 \dots dx_n \]
\[ = \int_{B_1 \times \dots \times B_n} \rho^{(n)}(x_1, \dots, x_n) \, dx_1 \dots dx_n. \]
We will denote equality in distribution by $ $.
Definition (Finite-dimensional distributions). The finite-dimensional distributions of a point process \(\Phi\) on \((E, \mathcal{B})\) are the joint probability distributions of the random vectors \[ \big( \Phi(B_1), \dots, \Phi(B_n) \big) \] for all \(n \in \mathbb{N}\) and \(B_1, \dots, B_n \in \mathcal{B}\).
Proposition. Let \(\Phi\) and \(\Phi'\) be point processes on \(E\).
If \[
\big( \Phi(B_1), \dots, \Phi(B_n) \big) \overset{d}{=} \big( \Phi'(B_1), \dots, \Phi'(B_n) \big)
\] for all \(B_1, \dots, B_n \in \mathcal{B}\), then \[
\Phi \overset{d}{=} \Phi'.
\]
Note: It actually suffices to take all \(B_1, \dots, B_n\) to be pairwise disjoint.
Definition (Laplace functional). The Laplace functional of a point process \(\Phi\) on \(E\) is defined by \[ \mathcal{L}_\Phi(f) := \mathbb{E}\left[ \exp \left( - \int_E f \, d\Phi \right) \right] \] for all measurable functions \(f: E \to \mathbb{R}_+\).
Proposition. If \(\Phi\) and \(\Phi'\) are point processes on \(E\) and \[ \mathcal{L}_\Phi(f) = \mathcal{L}_{\Phi'}(f) \] for all measurable \(f: E \to \mathbb{R}_+\), then \[ \Phi \overset{d}{=} \Phi'. \]
Proof. Let \(f = \sum_{i=1}^n c_i \mathbf{1}_{B_i}\) with \(c_i \in (0, \infty)\) and \(B_i \in \mathcal{B}\). Then \[ \mathcal{L}_\Phi(f) = \mathbb{E}\left[ \exp\left( - \sum_{i=1}^n c_i \Phi(B_i) \right) \right] = \mathcal{L}_{(\Phi(B_1), \dots, \Phi(B_n))}(c_1, \dots, c_n), \] which is precisely the multivariate Laplace transform of \((\Phi(B_1), \dots, \Phi(B_n))\).
Thus, \[ \mathcal{L}_\Phi(f) = \mathcal{L}_{\Phi'}(f) \quad \forall f \quad \Rightarrow \quad (\Phi(B_1), \dots, \Phi(B_n)) \overset{d}{=} (\Phi'(B_1), \dots, \Phi'(B_n)) \] and hence, by the previous proposition, \[ \Phi \overset{d}{=} \Phi'. \]
\(\square\)
Example (Binomial point process). Recall the binomial point process \[ \Phi = \sum_{i=1}^n \delta_{X_i}, \] where \(\{X_i\}_{i=1}^n\) are i.i.d. random elements of \(E\) with distribution \(\mathbb{Q}\).
The Laplace functional of \(\Phi\) is
\[\begin{align*} \mathcal{L}_\Phi(f) &= \mathbb{E}\left[ \exp\left( - \int_E f\, d\Phi \right) \right] = \mathbb{E}\left[ \exp\left( - \sum_{i=1}^n f(X_i) \right) \right] \\ &= \mathbb{E}\left[ \prod_{i=1}^n e^{-f(X_i)} \right] = \prod_{i=1}^n \mathbb{E}\left[ e^{-f(X_i)} \right] \quad (\text{i.i.d.}) \\ &= \left( \mathcal{L}_{f(X_1)}(1) \right)^n. \end{align*}\]
Definition (Void probability function). The void probability function of a point process \(\Phi\) is defined by \[ \gamma_\Phi(B) = \mathbb{P}[\Phi(B) = 0], \quad B \in \mathcal{B}. \]
Proposition. The void probability function \(\gamma_\Phi\) characterizes the distribution of a simple point process \(\Phi\).
Proof. See [baccelli2024random]{Thm.~2.1.10}.
\(\square\)
Example (Void probability of a binomial point process). Consider the binomial point process \[ \Phi = \sum_{i=1}^n \delta_{X_i}, \quad X_i \stackrel{\text{i.i.d.}}{\sim} \mathbb{Q}. \] Then the void probability is
\[\begin{align*} \mathbb{P}[\Phi(B) = 0] &= \mathbb{P}\left[ \bigcap_{i=1}^n \{ X_i \notin B \} \right] \\ &= \left[ 1 - Q(B) \right]^n. \end{align*}\]