Convergence in distribution of random compact sets in Polish spaces

Abstract

Let [phi],[phi]1,[phi]2,... be a sequence of random compact sets on a complete and separable metric space (S,d). We assume that P{[phi]n[intersection]B=[empty set]}-->P{[phi][intersection]B=[empty set]} for all B in some suitable class and show that this assumption determines if the sequence {[phi]n} converges in distribution to [phi]. This is an extension to general Polish spaces of the weak convergence theory for random closed sets on locally compact Polish spaces found in Norberg [1984. Convergence and existence of random set distributions. Ann. Probab. 12, 726-732.]Convergence in distribution Random compact sets Polish spaces