Question: How can we understand or describe the distribution of a point process \(\Phi\)?
We want to find analogues of classical distributional characteristics for random variables:
Setting:
Let \(E\) be a complete separable metric space (CSMS), \(\mathcal{B} := \mathcal{B}(E)\) be the Borel \(\sigma\)-algebra on \(E\), \(\mathcal{C}\) be the set of all compact subsets of \(E\), \((\mathcal{M}(E), \mathscr{M}(E))\) be the space of all counting measures on \(E\), and \(\Phi\) be a simple point process on \(E\).
Definition (First moment/intensity measure). The intensity measure \(\Lambda\) of \(\Phi\) is the measure on \((E, \mathcal{B})\) defined by \[ \Lambda(B) := \mathbb{E}[\Phi(B)], \qquad B \in \mathcal{B}(E). \] \(\Lambda(B)\) represents the expected number of points of \(\Phi\) in the set \(B\).
Even though \(\Phi\) is locally finite almost surely, it is not necessarily a locally finite measure. In other words, \[\forall C \in \mathcal{C},\ \mathbb{P}\big[ \Phi(C) < \infty \big] = 1 \nRightarrow \mathbb{E}\big[ \Phi(C) \big] < \infty.\]
In the future, we will need to assume this property.
Example (Binomial point process). If \(\Phi = \sum_{i=1}^n \delta_{X_i}\) is a binomial point process with \(X_i \stackrel{\text{i.i.d.}}{\sim} \mathbb{Q}\), then \[ \Lambda(B) = \mathbb{E}[\Phi(B)] = \mathbb{E}\left[\sum_{i=1}^n 1_{\{X_i \in B\}}\right] = \sum_{i=1}^n \mathbb{P}(X_i \in B) = n \mathbb{Q}(B). \]
Example (Mixed binomial point process).
Theorem (Campbell’s theorem). Let \(\Phi\) be a point process on \(E\) with intensity measure \(\Lambda\) and let \(f\) be a \(\Lambda\)-integrable/non-negative function. Then \[ \mathbb{E}\left[ \int_E f(x) \, \Phi(dx) \right] = \mathbb{E}\left[\sum_{x\in\Phi}f(x) \right] = \int_E f(x) \, \Lambda(dx). \]
Proof. Let \(f = 1_A\), where \(A \in \mathcal{B}\). Then \[ \mathbb{E}\left[ \int_E 1_A \, d\Phi \right] = \mathbb{E}\left[ \Phi(A) \right] = \Lambda(A) = \int_E 1_A \, d\Lambda. \]
By a standard argument in measure theory, we can build up a \(\Lambda\)-integrable function through simple functions and use the monotone convergence theorem.
\(\square\)