, 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
Let be a complete separable metric space. Let denote the set of all compact subsets of . The Hausdorff metric on is defined by
is also а complete separable metric space. The corresponding open subsets generate a σ-algebra on , the Borel sigma algebra of .
A random compact set is а measurable function from а probability space into .
Put another way, a random compact set is a measurable function such that is almost surely compact and
is a measurable function for every .
Random compact sets in this sense are also random closed sets as in Matheron (1975). Consequently their distribution is given by the probabilities
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
For , the probability is obtained, which satisfies
Thus the covering function is given by
Of course, can also be interpreted as the mean of the indicator function
The covering function takes values between and . The set of all with is called the support of . The set , of all with is called the kernel, the set of fixed points, or essential minimum . If , is а sequence of i.i.d. random compact sets, then almost surely
and converges almost surely to
- 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.