Definitions

Random compact set

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 $\left(M, d\right)$ be a complete separable metric space. Let $mathcal\left\{K\right\}$ denote the set of all compact subsets of $M$. The Hausdorff metric $h$ on $mathcal\left\{K\right\}$ is defined by

$h\left(K_\left\{1\right\}, K_\left\{2\right\}\right) := max left\left\{ sup_\left\{a in K_\left\{1\right\}\right\} inf_\left\{b in K_\left\{2\right\}\right\} d\left(a, b\right), sup_\left\{b in K_\left\{2\right\}\right\} inf_\left\{a in K_\left\{1\right\}\right\} d\left(a, b\right) right\right\}.$

$\left(mathcal\left\{K\right\}, h\right)$ is also а complete separable metric space. The corresponding open subsets generate a σ-algebra on $mathcal\left\{K\right\}$, the Borel sigma algebra $mathcal\left\{B\right\}\left(mathcal\left\{K\right\}\right)$ of $mathcal\left\{K\right\}$.

A random compact set is а measurable function $K$ from а probability space $\left(Omega, mathcal\left\{F\right\}, mathbb\left\{P\right\}\right)$ into $\left(mathcal\left\{K\right\}, mathcal\left\{B\right\} \left(mathcal\left\{K\right\}\right) \right)$.

Put another way, a random compact set is a measurable function $K : Omega to 2^\left\{Omega\right\}$ such that $K\left(omega\right)$ is almost surely compact and

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

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

$mathbb\left\{P\right\} \left(X cap K = emptyset\right)$ for $K in mathcal\left\{K\right\}.$

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 $mathbb\left\{P\right\}\left(X subset K\right).$

For $K = \left\{ x \right\}$, the probability $mathbb\left\{P\right\} \left(x in X\right)$ is obtained, which satisfies

$mathbb\left\{P\right\}\left(x in X\right) = 1 - mathbb\left\{P\right\}\left(x notin X\right).$

Thus the covering function $p_\left\{X\right\}$ is given by

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

Of course, $p_\left\{X\right\}$ can also be interpreted as the mean of the indicator function $mathbf\left\{1\right\}_\left\{X\right\}:$

$p_\left\{X\right\} \left(x\right) = mathbb\left\{E\right\} mathbf\left\{1\right\}_\left\{X\right\} \left(x\right).$

The covering function takes values between $0$ and $1$. The set $b_\left\{X\right\}$ of all $x in M$ with $p_\left\{X\right\} \left(x\right) > 0$ is called the support of $X$. The set $k_X$, of all $x in M$ with $p_X\left(x\right)=1$ is called the kernel, the set of fixed points, or essential minimum $e\left(X\right)$. If $X_1, X_2, ldots$, is а sequence of i.i.d. random compact sets, then almost surely

$bigcap_\left\{i=1\right\}^infty X_i = e\left(X\right)$

and $bigcap_\left\{i=1\right\}^infty X_i$ converges almost surely to $e\left(X\right).$

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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