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

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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

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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

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  • Matheron, G. (1975) Random Sets and Integral Geometry. J.Wiley & Sons, New York. ISBN 0-471-57621-2
  • Molchanov, I. (2005) The Theory of Random Sets. Springer, New York. ISBN 1-85233-892-X
  • Stoyan D., and H.Stoyan (1994) Fractals, Random Shapes and Point Fields. John Wiley & Sons, Chichester, New York. ISBN 0-471-93757-6