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.

[edit] 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 : \Omega \to 2^{M} such that K(ω) is almost surely compact and

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

is a measurable function for every x \in M.

[edit] 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 pX is given by

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

Of course, pX 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 bX of all x \in M with pX(x) > 0 is called the support of X. The set kX, of all x \in M with pX(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).

[edit] References

  • Matheron, G. (1975) Random Sets and Integral Geometry. J.Wiley & Sons, New York.
  • Stoyan D., and H.Stoyan (1994) Fractals, Random Shapes and Point Fields. John Wiley & Sons, Chichester, New York.
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