Strong law of large numbers for unions of random closed sets (Q2368163)

The distribution of a random closed set \(A\) in \(\mathbb{R}^ d\) (RACS) is determined by the corresponding (Choquet) capacity functional \(T(K)=P[A \cap K \neq \emptyset]\), where \(K \in {\mathcal K}\), the class of compact sets in \(\mathbb{R}^ d\) [basic definitions and properties concerning RACS are given by \textit{G. Matheron}, Random sets and integral geometry (1975; Zbl 0321.60009)]. According to the results of \textit{R. A. Davis}, \textit{E. Mulrow} and \textit{S. I. Resnick} [Adv. Appl. Probab. 20, No. 3, 573-599 (1988; Zbl 0656.60026)] a sequence \(\{X_ n\}\) of RACS converges a.s. as \(n \to \infty\) in the class \({\mathcal F}\) of closed sets in \(\mathbb{R}^ d\) to a nonrandom closed set \(X\), iff (i) for each \(K \in {\mathcal K}\), the relation \(K \cap X=\emptyset\) implies that \[ P[X_ n \cap K \neq \emptyset \text{ i.o.}] =P\left[\bigcap^ \infty_{n=1} \bigcup^ \infty_{m=n}\{X_ n \cap K \neq \emptyset\}\right]=0, \] and (ii) for any open set \(G \in {\mathcal G}\) in \(\mathbb{R}^ d\), if \(G \cap X \neq \emptyset\), then \[ P[X_ n \cap G=\emptyset \text{ i.o.}]=0. \] A class of sets \({\mathcal M}\) is said to determine \({\mathcal F}\)-convergence if the above result is valid after replacing \({\mathcal K}\) with \({\mathcal M}\) and \({\mathcal G}\) with \({\mathcal M}'=\{\text{Int} K:K \in {\mathcal M}\}\). Let \(\mathbb{S}^{d-1}\) be the unit sphere in \(\mathbb{R}^ d\) with the induced topology. Put \[ \begin{multlined} {\mathcal M}=\{ \{ux:u \in S, a \leq x \leq b\}:S \subset \mathbb{S}^{d-1}, \\ S \text{ coincides with the closure of its interior, } 0 \leq a<b\}, \end{multlined} \] \[ \hat K=\bigcup \{s\in K:s \geq 1\} \text{ for } K \in {\mathcal K}, \] and, for any functional \(\Lambda:{\mathcal M} \to [0,\infty]\), \[ Z(\Lambda; {\mathcal M})=\Biggl\{\bigcup \{\text{Int} F:F \in {\mathcal M},\Lambda (F)>1\}\Biggr\}^ c. \] The capacity \(R\) is called regularly varying on \({\mathcal M}\) with limit capacity \(\Lambda\) \((R \in\text{RV}(\beta,{\mathcal M},\Lambda,g))\) if for all \(F \in {\mathcal M}\), \(\lim_{t \to \infty} R(tF)/g(t)=\Lambda(F)\), where \(g:(0,\infty) \to (0,\infty)\) is a regularly varying function of index \(\beta>0\). The main result of the paper is the following Theorem 3.1. Let \(A_ 1,A_ 2,\dots\) be i.i.d. RACS with the capacity functional \(T\) and let \({\mathcal M}\) determine \({\mathcal F}\)-convergence. Define the capacity \(R(K)=- \log T(K)\), \(K \in {\mathcal K}\), with possibly infinite values. Assume that \(R \in\text{RV} (\beta,{\mathcal M},\Lambda,g)\) and \(R\) is strictly monotone on \({\mathcal M}\) (i.e. \(R(K_ 1)>R(K)\) for any \(K,K_ 1 \in {\mathcal M}\) such that \(K_ 1 \subset \text{Int} K\) and \(R(K)<\infty)\). Furthermore, suppose that \(\lim_{k \to \infty} R(t \hat K)/g(t)=\Lambda (\hat K)=\Lambda(K)\) and take \(\{a_ n\}\) satisfying \(g(a_ n) \simeq \log n\). Then, as \(n \to \infty\), \[ X_ n=a_ n^{-1} (A_ 1 \cup \cdots \cup A_ n) \to Z (\Lambda;{\mathcal M}) \text{ a.s. and conv} X_ n \to \text{conv} Z(\Lambda;{\mathcal M}) \text{ a.s. in } {\mathcal F}. \] Also the a.s. convergence of \(\{X_ n\}\) in the Hausdorff metric on \({\mathcal K}\) is considered and for special RACS certain results of \textit{R. A. Davis} et al. (loc. cit.) are obtained.