Random closed sets: results and problems
Ilya S. Molchanov · 2019
This chapter begins with distributions and examples of random sets. It discusses convergence of random set distributions, limits theorems for H. Minkowski sums and unions of random sets. The chapter outlines some problems in statistical inference. The mathematical theory of random closed sets was essentially laid by G. Matheron. Starting from G. Choquet’s theory of positively defined functions on cones, he characterized distributions of random sets and established many natural links to integral geometry and mathematical morphology. Random functions give rise to many examples of random sets that appear as graphs, hypo- or epi-graphs, and level sets of random functions. Set-valued random functions appear naturally in stochastic control problems. Epi-convergence of random functions is useful in the studies of extremal processes. Union-stable random sets may serve as a model of random closed sets, since the capacity functionals of union-stable random sets are expressed by explicit formulae and union-stable sets can be easily simulated as scaled unions of ‘simple’ random sets.