Random compact set

In mathematics, a random compact set is essentially a compact set-valued random variable. Random compact sets are useful in the study of attractors for random dynamical systems.

Definition
Let $$(M, d)$$ be a complete separable metric space. Let $$\mathcal{K}$$ denote the set of all compact subsets of $$M$$. The Hausdorff metric $$h$$ on $$\mathcal{K}$$ is defined by


 * $$h(K_{1}, K_{2}) := \max \left\{ \sup_{a \in K_{1}} \inf_{b \in K_{2}} d(a, b), \sup_{b \in K_{2}} \inf_{a \in K_{1}} d(a, b) \right\}.$$

$$(\mathcal{K}, h)$$ is also а complete separable metric space. The corresponding open subsets generate a &sigma;-algebra on $$\mathcal{K}$$, the Borel sigma algebra $$\mathcal{B}(\mathcal{K})$$ of $$\mathcal{K}$$.

A random compact set is а measurable function $$K$$ from а probability space $$(\Omega, \mathcal{F}, \mathbb{P})$$ into $$(\mathcal{K}, \mathcal{B} (\mathcal{K}) )$$.

Put another way, a random compact set is a measurable function $$K : \Omega \to 2^{\Omega}$$ such that $$K(\omega)$$ is almost surely compact and


 * $$\omega \mapsto \inf_{b \in K(\omega)} d(x, b)$$

is a measurable function for every $$x \in M$$.

Discussion
Random compact sets in this sense are also random closed sets as in Matheron (1975). Consequently their distribution is given by the probabilities


 * $$\mathbb{P} (X \cap K = \emptyset)$$ for $$K \in \mathcal{K}.$$

In passing, it should be noted that the distribution of а random compact convex set is also given by the system of all inclusion probabilities $$\mathbb{P}(X \subset K).$$

For $$K = \{ x \}$$, the probability $$\mathbb{P} (x \in X) $$ is obtained, which satisfies


 * $$\mathbb{P}(x \in X) = 1 - \mathbb{P}(x \not\in X).$$

Thus the covering function $$p_{X}$$ is given by


 * $$p_{X} (x) = \mathbb{P} (x \in X)$$ for $$x \in M.$$

Of course, $$p_{X}$$ can also be interpreted as the mean of the indicator function $$\mathbf{1}_{X}:$$


 * $$p_{X} (x) = \mathbb{E} \mathbf{1}_{X} (x).$$

The covering function takes values between $$ 0 $$ and $$ 1 $$. The set $$ b_{X} $$ of all $$x \in M$$ with $$ p_{X} (x) > 0 $$ is called the support of $$X$$. The set $$ k_X $$, of all $$ x \in M$$ with $$ p_X(x)=1 $$ is called the kernel, the set of fixed points, or essential minimum $$ e(X) $$. If $$ X_1, X_2, \ldots $$, is а sequence of i.i.d. random compact sets, then almost surely
 * $$ \bigcap_{i=1}^\infty X_i = e(X) $$

and $$ \bigcap_{i=1}^\infty X_i $$ converges almost surely to $$ e(X). $$