We can generalize the definition of a point process to a random locally finite measure.
Let \(\overline{\mathcal{M}}(\mathbb{R}^{d})\) be the space of locally finite measures on \(\mathbb{R}^{d}\), and
let \(\overline{\mathscr{M}}(\mathbb{R}^{d})\) be equipped with the smallest \(\sigma\)-algebra making the maps
\(\mu \mapsto \mu(B)\) measurable for all \(B \in \mathcal{B}(\mathbb{R}^{d})\).
Let \((\Omega, \mathcal{A}, \mathbb{P})\) be a probability space.
Definition. A random measure \(\Psi\) on \(\mathbb{R}^{d}\) is a random element of \(\overline{\mathcal{M}}(\mathbb{R}^{d})\),
i.e., a measurable map \(\Psi : \Omega \to \overline{\mathcal{M}}(\mathbb{R}^{d})\).
As with point processes, the distribution of a random measure \(\Psi\) is determined by its finite-dimensional distributions
\((\Psi(B_{1}), \ldots, \Psi(B_{n})), \forall B_{1}, \ldots, B_{n} \in \mathcal{B}(\mathbb{R}^{d})\),
and by its Laplace functional: \[
L_{\Psi}(f) = \mathbb{E}\Big[e^{-\int f\, d\Psi}\Big],
\qquad \text{for all measurable } f : \mathbb{R}^{d} \to \mathbb{R}_{+}.
\]
For \(\Lambda \in \overline{\mathcal{M}}(\mathbb{R}^{d})\), let \(\Phi_{\Lambda}\) denote a Poisson process with intensity measure \(\Lambda\).
Definition. Let \(\Psi\) be a random measure on \(\mathbb{R}^{d}\).
Then the mixture \(\Phi_{\Psi}\) is called a Cox process on \(\mathbb{R}^{d}\).
That is, conditioned on \(\Psi\), \(\Phi_{\Psi}\) is Poisson with intensity measure \(\Psi\).
Example. Let \(\Psi = \xi \Lambda\), where \(\Lambda \in \overline{\mathcal{M}}(\mathbb{R}^{d})\) and \(\xi\) is a non-negative random variable.
Then the Cox process \(\Phi_{\Psi}\) is called a mixed Poisson process.
Theorem. Let \(\Phi_{\Psi}\) and \(\Phi_{\Psi'}\) be Cox processes with directing measures \(\Psi\) and \(\Psi'\).
Then
\[
\Phi_{\Psi} \overset{d}{=} \Phi_{\Psi'}
\quad \text{iff} \quad
\Psi \overset{d}{=} \Psi'.
\]
Definition. A random locally finite measure \(\Psi\) is stationary if
\[
t_x \Psi \overset{d}{=} \Psi, \qquad \forall x \in \mathbb{R}^{d}.
\]
Proposition. Let \(\Phi_{\Psi}\) be a Cox process directed by \(\Psi\).
Then \(\Phi_{\Psi}\) is stationary if and only if \(\Psi\) is stationary.
Proof. [Idea of the proof] \(t_x \Phi_{\Psi}\) is a Cox process directed by \(t_x \Psi\).
By the previous theorem, \[
t_x \Phi_{\Psi} \overset{d}{=} \Phi_{\Psi}
\quad \Longleftrightarrow \quad
t_x \Psi \overset{d}{=} \Psi,
\] and thus, by the definition of stationarity,
\(\Phi_{\Psi}\) is stationary if and only if \(\Psi\) is stationary.
\(\square\)
A Cox process \(\Phi_{\Psi}\) with directing measure \(\Psi \in \mathcal{M}(\mathbb{R}^{d})\) has:
Remark: Recall that for a point process \(\Phi\) with intensity measure \(\Lambda\), \[ \mathrm{Var}(\Phi(B)) = \alpha^{(2)}(B \times B) - \Lambda(B)^2 + \Lambda(B). \]
For \(\Phi\) Poisson, \[ \mathrm{Var}(\Phi(B)) = \mathbb{E}[\Phi(B)]. \]
For the Cox process \(\Phi_{\Psi}\), \[ \mathrm{Var}(\Phi_{\Psi}(B)) = \mathbb{E}[\Psi(B)^2] - \mathbb{E}[\Psi(B)]^2 + \mathbb{E}[\Psi(B)] = \mathrm{Var}(\Psi(B)) + \mathbb{E}[\Psi(B)]. \]
Therefore, \[ \mathrm{Var}(\Phi_{\Psi}(B)) \ge \mathbb{E}[\Psi(B)]. \]
Hence, the number of points in a given set has greater variability in a Cox process than in a Poisson process.
Example (Cox Processes with Random Intensity Fields). Let \(\{Y(x)\}_{x \in \mathbb{R}^{d}}\) be a non-negative stochastic process on \(\mathbb{R}^{d}\).
Define the random measure by
$$
\Psi(B) = \int_{B} Y(x)\, dx, \qquad B \in \mathcal{B}(\mathbb{R}^{d}),
$$
assuming $Y$ is locally integrable a.s.
Then $\Phi_{\Psi}$ is a Cox process with random intensity field $\{Y(x)\}$.Example (Specific Example: Log-Gaussian Cox Processes). Let \(\{Z(x)\}_{x \in \mathbb{R}^{d}}\) be a Gaussian process on \(\mathbb{R}^{d}\).
That is, for all \(x_1, \ldots, x_n \in \mathbb{R}^{d}\), \[
(Z(x_1), \ldots, Z(x_n))
\] are Gaussian random vectors, and its distribution is determined by the mean function
\[
\mu(x) = \mathbb{E}[Z(x)]
\] and covariance function
\[
C(x,y) = \mathrm{Cov}(Z(x), Z(y)).
\] Now define the random measure \(\Psi\) by
\[
\Psi(B) = \int_{B} e^{Z(x)}\, dx, \qquad B \in \mathcal{B}(\mathbb{R}^{d}).
\]
The Cox process $\Phi_{\Psi}$ is called a *Log-Gaussian Cox Process (LGCP)*.
(Coerrolly, Møller, Waagepetersen, 2016) showed that for a stationary LGCP $\Phi$ with underlying Gaussian process
$\{Z(x)\}_{x \in \mathbb{R}^{d}}$, the reduced Palm version
$
\Phi_{\Psi}^{0,!} := \Phi^{0} - \delta_{0}
$
is a LGCP with underlying Gaussian process $\{Z_{0}(x)\}_{x \in \mathbb{R}^{d}}$, where
$
Z_{0}(x) = Z(x) + C(0,x),
$
and $C$ is the covariance function of $\{Z(x)\}_{x \in \mathbb{R}^{d}}$.