Random compact set

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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[edit]

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[edit]

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[edit]

  • 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.