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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

$h(K_1,K_2):=\max \{{\sup }_{a\in K_1}{\inf }_{b\in K_2}d(a,b),{\sup }_{b\in K_2}{\inf }_{a\in K_1}d(a,b)\}.$

$(\mathcal{𝒦},h)$ 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 $(\mathrm{\Omega },\mathcal{ℱ},\mathbb{\mathbb{P}})$ into $(\mathcal{𝒦},\mathcal{ℬ}(\mathcal{𝒦}))$.

Put another way, a random compact set is a measurable function $K:\mathrm{\Omega }\to 2^M$ 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

$\mathbb{\mathbb{P}}(X\cap K=\mathrm{\varnothing })$ for $K\in \mathcal{𝒦}.$

(The distribution of а random compact convex set is also given by the system of all inclusion probabilities $\mathbb{\mathbb{P}}(X\subset K).$)

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

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

Thus the covering function $p_X$ is given by

$p_X(x)=\mathbb{\mathbb{P}}(x\in X)$ 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 $X_1,X_2,\dots$, is а sequence of i.i.d. random compact sets, then almost surely

$\bigcap _{i=1}^{\mathrm{\infty }}{X}_i=e(X)$

and $\bigcap _{i=1}^{\mathrm{\infty }}{X}_i$ 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.