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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.