Recall from last time:
Consider a stationary \(k\)-flat process in \(\mathbb{R}^d\) with \(k = d-1\), i.e. a hyperplane process.
In this case, instead of working with a directional distribution \(\mathbb{Q}\) on \(\mathcal{G}(d,d-1)\), we can define a probability measure \(\varphi\) on the unit sphere \(S^{d-1}\) by: for \(A \in \mathcal{B}(S^{d-1})\), \[ \varphi(A) := \frac{1}{2}\, \mathbb{Q}\!\left(\{\, u^{\perp} : u \in A \,\}\right), \] where \(u^{\perp}:=\{x \in \mathbb{R}^d : \langle x, u\rangle = 0\}.\)
The measure \(\varphi\) is called the spherical directional distribution.
It satisfies \(\varphi(S^{d-1}) = 1,\) and \(\varphi(A) = \varphi(-A).\)
If \(\Phi\) is a stationary hyperplane process in \(\mathbb{R}^d\) with intensity \(\gamma\) and spherical directional distribution \(\varphi\), then \[ \int_{\mathcal{A}(d,d-1)} f\,d\Lambda = \gamma \int_{S^{d-1}} \int_{\mathbb{R}} f\bigl(H(u,t)\bigr)\,dt\, \varphi(du), \] where \[ H(u,t) := \{x \in \mathbb{R}^d : \langle u, x\rangle = t\}. \]
Example. If \(\Phi\) is isotropic (its distribution is invariant under rotations), then $ = _{d-1}, $ the normalized spherical Lebesgue measure.
Example. If $ = {i=1}^{n} ({u_i} + _{-u_i}), $ then the hyperplanes only have normal vectors \(u_1,\dots, u_n\).
Let $ {0} = {i } _{t_i} $ be a Poisson process on \(\mathbb{R}\) with intensity \(\gamma > 0\). Let \(\{u_i\}_{i \in \mathbb{Z}}\) be a collection of i.i.d. random unit vectors with distribution \(\varphi\), independent of \(\Phi_{0}\).
Then $ = {i } {H(u_i, t_i)} $ is a stationary Poisson hyperplane process with intensity \(\gamma\) and spherical directional distribution \(\varphi\).
To see this, note that $ = {i } {(t_i, u_i)} $ is an independent marking of \(\Phi_{0}\), and so is a Poisson process on \(\mathbb{R} \times S^{d-1}\) with intensity measure \(\gamma\,dt\,\varphi(du)\).
Then the map \((u,t) \mapsto H(u,t)\) is measurable, so the mapping theorem implies that \(\Phi\) is a Poisson process on \(\mathcal{A}(d,d-1)\).
Definition. For \(k \in \{2,\dots,d\}\), the hyperplanes \(H_1,\dots,H_k\) in \(\mathbb{R}^d\) are in general position if $ H_1 H_k $ is a \((d-k)\)-dimensional affine subspace.
Let \(\Phi\) be a stationary Poisson process in \(\mathbb{R}^d\) with intensity \(\gamma\) and spherical directional distribution \(\varphi\).
Fact: For any \((H_1,\dots,H_k) \in \Phi^{(k)}\), \[ H_1 \cap \cdots \cap H_k = \varnothing \quad\text{or}\quad H_1,\dots,H_k \text{ are in general position}. \]
Define the point process \(\Phi_{k}\) on \(\mathcal{A}(d,d-k)\) by \[ \Phi_{k}(A) := \frac{1}{k!} \sum_{(H_1,\dots,H_k)\in \Phi^{(k)}} \mathbf{1}_{\{H_1\cap\cdots\cap H_k \in A\}}. \]
Then \(\Phi_{k}\) is a stationary \((d-k)\)-flat process.
Question: Is \(\Phi_{k}\) a stationary Poisson flat process?
Answer: No. For example, there are almost surely no three collinear points in a stationary Poisson process.
Theorem. The intensity \(\gamma_{k}\) and directional distribution \(\mathbb{Q}_{k}\) of \(\Phi_{k}\) satisfy, for any \(A \in \mathcal{B}(\mathcal{G}(d,d-k))\), \[ \gamma_{k}\,\mathbb{Q}_{k}(A) = \frac{\gamma^{k}}{k!} \int_{S^{d-1}} \cdots \int_{S^{d-1}} \mathbf{1}_{A}(u_{1}^{\perp}\cap \cdots \cap u_{k}^{\perp})\, \nabla_{d}(u_{1},\dots,u_{k})\, \varphi(du_{1})\cdots\varphi(du_{k}), \] where \(\nabla_{d}(u_{1},\dots,u_{k})\) is the volume of the parallelepiped spanned by \(u_{1},\dots,u_{k}\).
For \(k=d\), \[ \gamma_{d} = \frac{\gamma^{d}}{d!} \int_{S^{d-1}} \cdots \int_{S^{d-1}} \nabla_{d}(u_{1},\dots,u_{d})\, \varphi(du_{1})\cdots\varphi(du_{d}). \]