Definitions

sequentially compact set

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 (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 : Omega to 2^{Omega} 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}.

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{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 notin 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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