Characterization of universal classes in the Glivenko-Cantelli theorem for random closed sets. (Russian) Zbl 0696.60015

Molchanov, I. S.

Teor. Veroyatn. Mat. Stat., Kiev 41, 74-78 (1989).

Let \(A_ 1,...,A_ n\) be iid random sets with the common capacity functional \(T(K)=P\{A_ 1\cap K\neq \emptyset \}\), where K describes the class of all compact subsets of the basic space E, and let \[ T^*_ n(K)=n^{-1}\sum^{n}_{i=1}1_{A_ i\cap K\neq \emptyset} \] be the empirical capacity functional. A certain family of compacts \({\mathfrak M}\) is said to be a universal class if for every compact \(K_ 0\) \(\sup_{K\subset K_ 0,K\in {\mathfrak M}}| T^*_ n(K)-T(K)| \to 0\quad a.s.,\quad as\quad n\to \infty,\)regardless of the distribution of \(A_ 1\). It is proven that \({\mathfrak M}\) is a universal class iff for any compact K there exists such an \(n\geq 2\) that for every n-element family \(\{K_ 1,...,K_ n\}\subset {\mathfrak M}\), \(K_ i\subset K_ 0\), \(1\leq i\leq n\), one of the sets \(K_ 1,...,K_ n\) is covered by others. The proof uses the definition and properties of the Vapnik-Chervonenkis classes.

Reviewer: I.Molchanov

MSC:

60D05	Geometric probability and stochastic geometry
62G30	Order statistics; empirical distribution functions
54C60	Set-valued maps in general topology

Keywords:

stochastic geometry; law of large numbers; random sets; capacity functional; empirical capacity functional; Vapnik-Chervonenkis classes