Discussion
Random compact sets in this sense are also random closed sets as in Matheron (1975). Consequently 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
Read more about this topic: Random Compact Set
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