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| In [[mathematics]], a '''random compact set''' is essentially a [[compact space|compact set]]-valued [[random variable]]. Random compact sets are useful in the study of attractors for [[random dynamical system]]s.
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| ==Definition==
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| Let <math>(M, d)</math> be a [[complete space|complete]] [[separable space|separable]] [[metric space]]. Let <math>\mathcal{K}</math> denote the set of all compact subsets of <math>M</math>. The Hausdorff metric <math>h</math> on <math>\mathcal{K}</math> is defined by
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| :<math>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\}.</math>
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| <math>(\mathcal{K}, h)</math> is also а complete separable metric space. The corresponding open subsets generate a [[sigma algebra|σ-algebra]] on <math>\mathcal{K}</math>, the [[Borel sigma algebra]] <math>\mathcal{B}(\mathcal{K})</math> of <math>\mathcal{K}</math>.
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| A '''random compact set''' is а [[measurable function]] <math>K</math> from а [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> into <math>(\mathcal{K}, \mathcal{B} (\mathcal{K}) )</math>.
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| Put another way, a random compact set is a measurable function <math>K \colon \Omega \to 2^{M}</math> such that <math>K(\omega)</math> is [[almost surely]] compact and
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| :<math>\omega \mapsto \inf_{b \in K(\omega)} d(x, b)</math>
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| is a measurable function for every <math>x \in M</math>.
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| ==Discussion==
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| Random compact sets in this sense are also [[random closed set]]s as in Matheron (1975). Consequently their distribution is given by the probabilities
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| :<math>\mathbb{P} (X \cap K = \emptyset)</math> for <math>K \in \mathcal{K}.</math>
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| (The distribution of а random compact convex set is also given by the system of all inclusion probabilities <math>\mathbb{P}(X \subset K).</math>)
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| For <math>K = \{ x \}</math>, the probability <math>\mathbb{P} (x \in X) </math> is obtained, which satisfies
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| :<math>\mathbb{P}(x \in X) = 1 - \mathbb{P}(x \not\in X).</math>
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| Thus the '''covering function''' <math>p_{X}</math> is given by
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| :<math>p_{X} (x) = \mathbb{P} (x \in X)</math> for <math>x \in M.</math> | |
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| Of course, <math>p_{X}</math> can also be interpreted as the mean of the indicator function <math>\mathbf{1}_{X}</math>:
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| :<math>p_{X} (x) = \mathbb{E} \mathbf{1}_{X} (x).</math>
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| The covering function takes values between <math> 0 </math> and <math> 1 </math>. The set <math> b_{X} </math> of all <math>x \in M</math> with <math> p_{X} (x) > 0 </math> is called the '''support''' of <math>X</math>. The set <math> k_X </math>, of all <math> x \in M</math> with <math> p_X(x)=1 </math> is called the '''kernel''', the set of '''fixed points''', or '''essential minimum''' <math> e(X) </math>. If <math> X_1, X_2, \ldots </math>, is а sequence of [[i.i.d.]] random compact sets, then almost surely
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| :<math> \bigcap_{i=1}^\infty X_i = e(X) </math>
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| and <math> \bigcap_{i=1}^\infty X_i </math> converges almost surely to <math> e(X). </math>
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| == References ==
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| * Matheron, G. (1975) ''Random Sets and Integral Geometry''. J.Wiley & Sons, New York.
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| * Molchanov, I. (2005) ''The Theory of Random Sets''. Springer, New York.
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| * Stoyan D., and H.Stoyan (1994) ''Fractals, Random Shapes and Point Fields''. John Wiley & Sons, Chichester, New York.
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| [[Category:Random dynamical systems]]
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| [[Category:Probability theory]]
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| [[Category:Randomness]]
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