Random compact set

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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[edit]

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 .

Discussion[edit]

Random compact sets in this sense are also random closed sets as in Matheron (1975). Consequently, under the additional assumption that the carrier space is locally compact, their distribution is given by the probabilities

for

(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

for

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

References[edit]

  • 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.