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 σ-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 $K \colon \Omega \to 2^{M}$ 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}.

(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 $e(X).$

References

Matheron, G. (1975) Random Sets and Integral Geometry. J.Wiley & Sons, New York.
Molchanov, I. (2005) The Theory of Random Sets. Springer, New York.
Stoyan D., and H.Stoyan (1994) Fractals, Random Shapes and Point Fields. John Wiley & Sons, Chichester, New York.