Convergence of \({\mathcal S}'\)-valued processes and space-time random fields
The mapping \(x\to \tilde x\) of the Skorokhod space \(D([0,T]\), \({\mathcal S}'(R^ d))\) into \({\mathcal S}'(R^{d+1})\) defined by \[ <\tilde x,\Phi >= \int^{T}_{0}<x(t),\Phi (t,\cdot)>dt,\quad \Phi \in {\mathcal S}(R^{d+1}) \] is shown to be continuous. From this fact it follows that for a tight sequence of processes \((X_ n)\) in \(D([0,T]\), \({\mathcal S}'(R^ d))\), convergence indistribution of \({\mathcal S}'(R^{d+1})\)- valued random variables \((\tilde X_ n)\) implies weak convergence of \((X_ n)\). This establishes a new method of identifying the limit process. The method turns out to be convenient since nuclearity of \({\mathcal S}'(R^{d+1})\) implies possibility of using the Lévy continuity theorem. When \([0,T]\) is replaced by \([0,\infty)\) the space \({\mathcal S}'(R^{d+1})\) is not appropriate in general, and the question is investigated of what is the smallest nuclear space of distributions containing \({\mathcal S}'(R^{d+1})\) that can be used. These results are related to the asymptotic analysis of particle systems.
scientific article; zbMATH DE number 3946902
scientific article