Convergence in distribution of random compact sets in Polish spaces (Q2483448)

Let \(\varphi,\varphi_1,\varphi_2,\dots\) be a sequence of random compact sets on a complete and separable metric space \((S,d)\). We assume that \(P\{\varphi_n\cap B=\emptyset\}\to P\{\varphi\cap B=\emptyset\}\) for all \(B\) in some suitable class \(\mathcal B\) and show that this assumption determines if the sequence \(\{\varphi_n\}\) converges in distribution to \(\varphi\). This is an extension to general Polish spaces of the weak convergence theory for random closed sets on locally compact Polish spaces found in \textit{T. Norberg} [Ann. Probab. 12, 726--732 (1984; Zbl 0545.60021)].