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2023-06-17T20:06:57Z

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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.

==Definition==

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

:<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>

<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>.

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>.

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

:<math>\omega \mapsto \inf_{b \in K(\omega)} d(x, b)</math>

is a measurable function for every <math>x \in M</math>.

==Discussion==

Random compact sets in this sense are also [[random closed set]]s as in [[Georges Matheron|Matheron]] (1975). Consequently, under the additional assumption that the carrier space is locally compact, their distribution is given by the probabilities

:<math>\mathbb{P} (X \cap K = \emptyset)</math> for <math>K \in \mathcal{K}.</math>

(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>)

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

:<math>\mathbb{P}(x \in X) = 1 - \mathbb{P}(x \not\in X).</math>

Thus the '''covering function''' <math>p_{X}</math> is given by

:<math>p_{X} (x) = \mathbb{P} (x \in X)</math> for <math>x \in M.</math>

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

:<math>p_{X} (x) = \mathbb{E} \mathbf{1}_{X} (x).</math>

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

:<math> \bigcap_{i=1}^\infty X_i = e(X) </math>

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

==References==

* Matheron, G. (1975) ''Random Sets and Integral Geometry''. J.Wiley & Sons, New York. {{ISBN|0-471-57621-2}}
* Molchanov, I. (2005) ''The Theory of Random Sets''. Springer, New York. {{ISBN|1-85233-892-X}}
* Stoyan D., and H.Stoyan (1994) ''Fractals, Random Shapes and Point Fields''. John Wiley & Sons, Chichester, New York. {{ISBN|0-471-93757-6}}

{{Measure theory}}

[[Category:Random dynamical systems]]
[[Category:Statistical randomness]]

Random compact set - Revision history

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