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
![{\displaystyle h(K_{1},K_{2}):=\max \left\{\sup _{a\in K_{1}}\inf _{b\in K_{2}}d(a,b),\sup _{b\in K_{2}}\inf _{a\in K_{1}}d(a,b)\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31cb6e7540ed1218a3cb7557b1273a2365fd1bfb)
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
![{\displaystyle \omega \mapsto \inf _{b\in K(\omega )}d(x,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/658f5b388674d664aaecea6fc8d58a884c225127)
is a measurable function for every
.
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 ![{\displaystyle K\in {\mathcal {K}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8983ece62573986aec506ed44fb9480f75620d8b)
(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
![{\displaystyle \mathbb {P} (x\in X)=1-\mathbb {P} (x\not \in X).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a5facb7e16e525034be1565569c0bf86c9e696)
Thus the covering function
is given by
for ![{\displaystyle x\in M.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/127645d1eb7572c38f34e5b855f5d789404f6d93)
Of course,
can also be interpreted as the mean of the indicator function
:
![{\displaystyle p_{X}(x)=\mathbb {E} \mathbf {1} _{X}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a0eab12fd946935384d3c41d4212b2a27d0f1a)
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
![{\displaystyle \bigcap _{i=1}^{\infty }X_{i}=e(X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47ec9449e8c7364bb4b32ed29bb6cb8db963f1ec)
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.