Recall Campbell’s Theorem: for a stationary point process \(\Phi\) on \(\mathbb{R}^d\) with intensity \(\lambda \in (0,\infty)\), [ = _{^d} f(x), dx. ]
Related to the nearest-neighbor question: in a set \(B \subseteq \mathbb{R}^d\),
what is the expected number of points of \(\Phi\) such that the closest neighbor is farther than distance \(r\)?
\[ \mathbb{E}\left[ \sum_{k=1}^{\infty} \mathbf{1}_{\{X_k \in B\}} \, \mathbf{1}_{\{\Phi(B_d(X_k, r)) = 1\}} \right], \] where \(B_d(X_k, r) = \{x \in \mathbb{R}^d : \|x - X_k\| \le r\}.\)
Definition. The Campbell measure of a point process \(\Phi\) on \(\mathbb{R}^d\) is the measure
\(\mathcal{C}\) on \(\mathbb{R}^d \times \mathcal{M}(\mathbb{R}^d)\) defined by \[
\mathcal{C}(B \times A)
=
\mathbb{E}\left[\Phi(B)\, \mathbf{1}_{\{\Phi \in A\}}\right],
\qquad
A \in \mathscr{M}(\mathbb{R}^d),\;
B \in \mathcal{B}(\mathbb{R}^d).
\]
Note: \(\mathcal{C}(B \times \mathcal{M}(\mathbb{R}^d)) = \Lambda(B).\)
Then, for measurable functions
\(f : \mathbb{R}^d \times \mathcal{M}(\mathbb{R}^d) \rightarrow \mathbb{R}_{+}\), \[
\mathbb{E}\left[ \int_{\mathbb{R}^d} f(x, \Phi)\, \Phi(dx) \right]
=
\int_{\mathbb{R}^d \times \mathcal{M}(\mathbb{R}^d)}
f(x, \nu)\, \mathcal{C}(d(x,\nu)).
\]
We can decompose \(\mathcal{C}\) in the stationary case and obtain the following result:
Theorem (Refined Campbell’s Theorem). Let \(\Phi\) be a stationary poisson process on \(\mathbb{R}^d\) with intensity \(\lambda \in (0,\infty)\).
Then there is a unique probability measure \(\mathbb{P}_{\Phi}^{0}\) on \(\left(\mathcal{M}(\mathbb{R}^d), \mathscr{M}(\mathbb{R}^d)\right)\) such that
for all measurable \(f : \mathbb{R}^d \times \mathcal{M}(\mathbb{R}^d) \rightarrow \mathbb{R}_{+}\),
\[ \mathbb{E}\left[ \int_{\mathbb{R}^d} f(x, t_x \Phi)\, \Phi(dx) \right] = \lambda \int_{\mathbb{R}^d} \int_{\mathcal{M}(\mathbb{R}^d)} f(x, \nu)\, P_{\Phi}^{0}(d\nu)\, dx, \] or
\[ \mathbb{E}\left[ \int_{\mathbb{R}^d} f(x, \Phi)\, \Phi(dx) \right] = \lambda \int_{\mathbb{R}^d} \int_{\mathcal{M}(\mathbb{R}^d)} f(x, t_{-x} \nu)\, P_{\Phi}^{0}(d\nu)\, dx. \]
Corollary. \(P_{\Phi}^{0}(A) = \frac{1}{\lambda |B|} \mathbb{E}\left[ \int_{\mathbb{R}^d}\mathbf{1}_{B}(x) \mathbf{1}_{A}(t_x \Phi)\, \Phi(dx) \right], \forall A \in \mathscr{M}(\mathbb{R}^d).\)
Note: Since \(\mathbf{1}_{\{t_x \Phi(\{0\}) > 0\}}=\mathbf{1}_{\{\Phi(\{x\}) > 0\}}\), we obtain
\[\begin{align*} &P_{\Phi}^{0}\left(\{\nu \in \mathcal{M}(\mathbb{R}^d) : \nu(\{0\}) > 0\}\right) \\ =& \frac{1}{\lambda}\mathbb{E}\left[\int_{\mathbb{R}^d}\mathbf{1}_{[0,1]^d}(x)\mathbf{1}_{\{\Phi(\{x\}) > 0\}}\, \Phi(dx)\right]\\ =&\frac{1}{\lambda} \mathbb{E}\left[ \Phi([0,1]^d) \right] \\ =&\frac{1}{\lambda} \lambda |[0,1]^d|\\ =&1. \end{align*}\]
That is, \(P_{\Phi}^{0}\) has support on \(\nu \in \mathcal{M}(\mathbb{R}^d)\) with an atom at the origin.
Definition. \(P_{\Phi}^{0}\) is called the Palm distribution of \(\Phi\).
The point process \(\Phi^{0}\) with distribution \(P_{\Phi}^{0}\) is called the Palm version of \(\Phi\).
Definition. Let \(\Phi^{0}\) be the Palm version of a stationary Poisson process \(\Phi\).
Then \(\Phi_{\Phi}^{0,!} := \Phi^{0} - \delta_{0}\) is called the reduced Palm version of \(\Phi\).
Theorem (Slivnyak’s Theorem). Let \(\Phi\) be a stationary point process on \(\mathbb{R}^{d}\).
Then \(\Phi\) is Poisson if and only if
\[
P_{\Phi}^{0}(A) = \mathbb{P}(\Phi + \delta_{0} \in A),
\] if and only if ,
\[
P_{\Phi}^{0,!}(A) = \mathbb{P}(\Phi \in A).
\]
Definition. The nearest neighbor distribution of a stationary Poisson process on \(\mathbb{R}^{d}\) is
\[
D(r) := P_{\Phi}^{0,!}\big(\Phi(B_{d}(0,r)) > 0\big).
\]
Example. Nearest neighbor distribution of a stationary Poisson process on \(\mathbb{R}^{d}\) with intensity \(\lambda\): \[ D(r) = P_{\Phi}^{0,!}\big(\Phi(B_{d}(0,r)) > 0\big) = \mathbb{P}\big(\Phi(B_{d}(0,r)) > 0\big) = 1 - e^{-\lambda |B_{d}(0,r)|}. \]
Exercise (Matérn I Hard-Core Point Process). Let \(\Phi = \sum_{k \in \mathbb{N}} \delta_{X_k}\) be a stationary Poisson process on \(\mathbb{R}^{d}\) with intensity \(\lambda \in (0,\infty)\).
Define
$
\widetilde{\Phi}
=
\sum_{k \in \mathbb{N}}
\delta_{X_k}\,
\mathbf{1}_{\{\Phi(B_{d}(X_k,r)) = 1\}}.
$
**Question:** What is the intensity measure of $\widetilde{\Phi}$?