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

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.

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