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 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
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
- 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.