Let \(\Phi\) be a Poisson process on \(\mathbb{R}^d\) with intensity measure \[ \Lambda(\cdot) = \lambda |\cdot|, \] where \(|\cdot|\) denotes the Lebesgue measure. Then \(\Phi\) is called a homogeneous Poisson process with intensity \(\lambda\).
To simulate a realization of \(\Phi\) on a compact window \(W \subset \mathbb{R}^d\), proceed as follows:
Formally, the distribution of each point is \[ X_i \sim \frac{\Lambda(\cdot \cap W)}{\Lambda(W)} = \frac{\lambda |\cdot \cap W|}{\lambda |W|} = \frac{|\cdot \cap W|}{|W|}. \]
Let \((E, \mathcal{B})\) be a locally compact complete separable metric space (CSMS) with Borel \(\sigma\)-algebra \(\mathcal{B}\).
Recall that for a point process \(\Phi\) on \(E\), we can write $ = {k=1}^N {X_k}, $ where \(N = \Phi(E)\) is a random variable taking values in \(\mathbb{N} \cup \{\infty\}\).
Definition (Thinning). Let $ = {k=1}^N {X_k} $ be a point process on \(E\), and let \(p: E \to [0,1]\) be a measurable function.
Let \(\{U_k\}_{k\geq 1}\) be a sequence of i.i.d. \(\mathrm{Uniform}[0,1]\) random variables independent of \(\Phi\).
Then the \(p\)-thinning of \(\Phi\) is the point process \[ \widetilde{\Phi} = \sum_{k=1}^N \mathbf{1}\{ U_k \leq p(X_k) \} \, \delta_{X_k}. \]
Proposition (Laplace functional under thinning). Let \(\widetilde{\Phi}\) be a \(p\)-thinning of $ = {k=1}^N {X_k}, $ and suppose \(\Phi\) has intensity measure \(\Lambda\).
Then the Laplace functional of \(\widetilde{\Phi}\) satisfies \[
L_{\widetilde{\Phi}}(f) = L_{\Phi}(\tilde{f}),
\] where \[
\tilde{f}(x) = -\log\left[ 1 - p(x) \big( 1 - e^{-f(x)} \big) \right]
\] for all measurable \(f: E \to \mathbb{R}_+\).
Proof. \[\begin{align*} L_{\widetilde{\Phi}}(f) &:= \mathbb{E}\left[ e^{ -\int_E f \, d\widetilde{\Phi} } \right] \\ &= \mathbb{E}\left[ e^{ - \sum_{k=1}^N \mathbf{1}\{ U_k \leq p(X_k) \} f(X_k) } \right] \\ &= \mathbb{E}\left[ \mathbb{E}\left[ e^{ - \sum_{k=1}^N \mathbf{1}\{ U_k \leq p(X_k) \} f(X_k) } \,\middle|\, \Phi \right] \right] \\ &= \mathbb{E}\left[ \prod_{k=1}^N \mathbb{E}\left[ e^{ - \mathbf{1}\{ U_k \leq p(X_k) \} f(X_k) } \,\middle|\, \Phi \right] \right] \\ &= \mathbb{E}\left[ \prod_{k=1}^N \Big( (1 - p(X_k)) + p(X_k) e^{-f(X_k)} \Big) \right] \\ &= \mathbb{E}\left[ e^{ - \int_E \tilde{f}(x) \, d\Phi(x) } \right] \\ &= L_{\Phi}(\tilde{f}). \end{align*}\]
\(\square\)
Corollary (Thinning of a Poisson process). The \(p\)-thinning of a Poisson process on \(E\) with intensity measure \(\Lambda\) is itself a Poisson process with intensity measure \[ \widetilde{\Lambda}(dx) = p(x)\Lambda(dx). \]
Proof. This follows directly from the Laplace functional characterization of a Poisson process and the previous proposition.
\(\square\)
Example (Simulation of an inhomogeneous Poisson process on \(\mathbb{R}^d\)). Let \(\Phi\) be a Poisson process on \(\mathbb{R}^d\) with intensity measure \[ \Lambda(dx) = \lambda(x)\, dx, \] where \(\lambda(x)\) is called the intensity function.
Assume \(\lambda(x) \leq \lambda^\star < \infty\).
We can simulate a realization of this inhomogeneous Poisson process as follows:
The simulated Poisson process has intensity measure \[ \Lambda(B) = \int_B \frac{\lambda(x)}{\lambda^\star} \lambda^\star dx = \int_B \lambda(x) dx. \]
Note: The intensity measure \(\widetilde{\Lambda}\) of the \(p\)-thinning \(\widetilde{\Phi}\) of \(\Phi\) with intensity measure \(\Lambda\) is given by \[ \widetilde{\Lambda}(B) = \int_B p(x)\, \Lambda(dx). \]
Let \((E, \mathcal{B}(E))\) and \((G, \mathcal{B}(G))\) be locally compact CSMS equipped with Borel \(\sigma\)-algebras. Let \(\mathcal{C}(E)\) and \(\mathcal{C}(G)\) denote the collections of compact sets of \(E\) and \(G\), respectively. Let \(T: E \rightarrow G\) be a measurable mapping and assume that \[ T^{-1}(C) \in \mathcal{C}(E) \quad \text{for all } C \in \mathcal{C}(G). \]
Then, for any locally finite measure \(\mu\) on \(E\), the image of \(\mu\) under \(T\) (also called the pushforward of \(\mu\)), defined by \[ T(\mu) = \mu \circ T^{-1}, \] is a locally finite measure on \(G\).
Proposition. In the above setting, if \(\Phi\) is a Poisson process on \(E\) with intensity measure \(\Lambda\), then \(T(\Phi)\) is a Poisson process with intensity measure \[ T(\Lambda) = \Lambda \circ T^{-1}. \]
Example. Let \(\Phi\) be a Poisson process on \(\mathbb{R}^d \setminus \{0\}\) with intensity measure [ (dx) = |x|^{-d} dx. ] Define the mapping [ T: ^d {0} (d, d-1) ] where \(\mathcal{A}(d, d-1)\) denotes the space of \((d-1)\)-dimensional affine subspaces of \(\mathbb{R}^d\). For \(x \in \mathbb{R}^d \setminus \{0\}\), set [ H(x) := { y ^d : x, y = 1 }. ]