On the existence of ordered couplings of random sets — with applications

AbstractLetψ andϕ be two given random closed sets in a locally compact second countable topological spaceS. (They need not be based on the same probability space.) The main result gives necessary and sufficient conditions on the distributions ofψ andϕ, for the existence of two random closed sets $$\hat \psi $$ and $$\hat \varphi $$ , based on the same probability space and such that their distributions coincide with those ofψ andϕ, resp., and $$\hat \psi \subseteq \hat \varphi $$ a.s.This coupling result tells us in particular when a probability distribution onS is selectionable w.r.t. (the distribution of) a random closed set. An existence result for realizable thinnings of a simple point process is obtained by specializing it to supports of random measures.The coupling result is extended to random variables in a countably based continuous poset. As examples we mention various kinds of random capacities — in particular random measures — and random compact (saturated) sets. Moreover, the extended result tells us when a probability distribution onS is selectionable w.r.t. the distribution of a random compact (saturated) set.

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