Choquet-Kendall-Matheron theorem
A theorem characterizing the distribution of a random closed set in terms of the Choquet capacity functional [a1]. This theorem was established independently by D.G. Kendall [a2] and G. Matheron [a3] in their work on random closed sets.
Let and
be the family of closed and compact subsets of
, respectively. The family
is endowed with the hit-or-miss topology, which is generated by
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for a compact set and open sets
. Let
be the Borel
-algebra generated by the hit-or-miss topology. Then
is the smallest
-algebra of subsets of
containing the sets
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Now, a random closed set is an
-valued random element. Its distribution is described by the corresponding probability measure
on
:
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This distribution can also be characterized by the functional
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The functional is an alternating Choquet capacity of infinite order. This means that:
i) is upper semi-continuous (
implies
; cf. also Semi-continuous function);
ii) ,
, where
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The values can be interpreted as the probability that
does not intersect
but does intersect
.
The Choquet–Kendall–Matheron theorem asserts that given a functional on
, there exists a distribution
on
with
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if and only if is an alternating Choquet capacity of infinite order with
and
. This distribution is necessarily unique.
See [a4] for applications.
References
[a1] | G. Choquet, "Theory of capacities" Ann. Inst. Fourier , V (1953-1954) pp. 131–295 |
[a2] | D.G. Kendall, "Foundations of a theory of random sets" E.F. Harding (ed.) D.G. Kendall (ed.) , Stochastic Geometry , Wiley (1974) pp. 322–376 |
[a3] | G. Matheron, "Random sets and integral geometry" , Wiley (1975) |
[a4] | D. Stoyan, W.S. Kendall, J. Mecke, "Stochastic geometry and its applications" , Wiley (1995) (Edition: Second) |
Choquet-Kendall-Matheron theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Choquet-Kendall-Matheron_theorem&oldid=46339