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Lecture 16: Random Closed Sets (Continued)

Introduction

Last time:

We defined the Fell topology on $\mathcal{F}$, the space of closed subsets of $\mathbb{R}^d$, and the Borel $\sigma$-algebra $\mathcal{B}(\mathcal{F})$.
We defined random closed sets (RACS) as measurable maps $Z:\Omega\to\mathcal{F}$.

Today:

Characterize the distribution of a RACS.
Consider stationary RACS and contact distributions.

Let $\mathcal{C}$ denote the set of compact subsets of $\mathbb{R}^d$.

Capacity Functional

We can characterize the distribution of a RACS with a functional haveing properties analogous to the distribution function of a random variable.

Definition. Let $Z$ be a RACS.
Its capacity functional is the mapping \[ T_Z : \mathcal{C}\to \mathbb{R}_+, \qquad T_Z(C) := \mathbb{P}(Z\in \mathcal{F}_C) = \mathbb{P}(Z\cap C\neq\varnothing). \]

Theorem. If $Z$ and $Z'$ are RACS such that $T_Z = T_{Z'}$, then $Z \stackrel{d}{=} Z'$.

Proof. The class $\{\mathcal{F}_C : C\in\mathcal{C}\}$ is nonempty, stable under finite intersections, and generates the Fell Borel $\sigma$-algebra.
If $Z$ and $Z'$ agree on this generating class, they agree on $\mathcal{B}(\mathcal{F})$.

$\square$

Example. Let $\Phi$ be a point process on $\mathbb{R}^d$, and let $Z = \mathrm{Supp}(\Phi)$.
Then \[ T_Z(C) = \mathbb{P}(\Phi(C) > 0) = 1 - \mathbb{P}(\Phi(C)=0) = 1 - v_\Phi(C), \] where $v_\Phi(C)$ is the void probability of $\Phi$.
For simple point processes, the distribution is completely determined by the void probabilities.

Properties of Capacity Functionals

For a RACS $Z$:

[(i)] $0 \le T_Z(C) \le 1$ for all $C\in\mathcal{C}$, and $T_Z(\varnothing)=0$.
[(ii)] Upper semicontinuity: If $C_n\downarrow C$, then \[ T_Z(C_n) \downarrow T_Z(C). \]
[(iii)] Complete alternation: for all $C_0,\ldots,C_k\in\mathcal{C}$, define recursively \[ S_0(C_0) := 1 - T_Z(C_0), \] \[ S_k(C_0;\,C_1,\ldots,C_k) := S_{k-1}(C_0;\,C_1,\ldots,C_{k-1}) - S_{k-1}(C_0\cup C_k;\,C_1,\ldots,C_{k-1}). \] Then \[ S_k(C_0;\,C_1,\ldots,C_k) \ge 0. \]

Theorem (Choquet). If $T:\mathcal{C}\to\mathbb{R}_+$ satisfies (i)–(iii), then there exists a RACS $Z$ with $T_Z=T$.

Stationary RACS

Definition. A RACS $Z$ is stationary if for all $x\in\mathbb{R}^d$, $ Z Z - x. $

By the capacity functional characterization, $Z$ is stationary iff \[ T_Z(C) = T_Z(C - x) \qquad \forall C\in\mathcal{C},\; x\in\mathbb{R}^d. \]

For any RACS $Z$, consider the stochastic process on $\mathbb{R}^d$ given by $ Z(x)={{xZ}}. $

Define the mean and covariance functions: \[ m_Z(x) := \mathbb{E}[\mathbf{1}_Z(x)] = \mathbb{P}(x\in Z), \] \[ k_Z(x,y) := \mathbb{E}[(\mathbf{1}_Z(x)-m_Z(x))(\mathbf{1}_Z(y)-m_Z(y))] = \mathbb{P}(x,y\in Z) - m_Z(x)m_Z(y). \]

For stationary RACS:

$m_Z(x)=p:=\mathbb{P}(0\in Z)$, the volume fraction,
$k_Z(x,y)=k_Z(x-y,0)$,
$k_Z(x,0)=\mathbb{P}(0,x\in Z) - p^2$.

Another way to describe stationary RACS is by contact distributions.

Definition. Let $K\subset\mathbb{R}^d$ be a compact convex body containing the origin.
Define the $K$-distance of $x\in\mathbb{R}^d$ from $F\in\mathcal{F}'$ by \[ d_K(x,F) := \min\{ r>0 : x + rK \cap F \neq \varnothing \}. \]

Definition. The contact distribution of a RACS $Z$ w.r.t. $K$ is \[ H_K(r) := \mathbb{P}( d_K(0,Z) \le r \mid 0\notin Z ), \qquad r\ge 0. \]

Special Cases:

$K=\mathbb{B}^d$: spherical contact distribution.
$K=[0,u]$: linear contact distribution, measuring distance in direction $u$.

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