Convergence of \({\mathcal S}'\)-valued processes and space-time random fields
type: Item
instance of: scholarly article
title: Convergence of \({\mathcal S}'\)-valued processes and space-time random fields
zbMATH Open document ID: 0589.46027
DOI: 10.1016/0022-1236(86)90078-9
author: B. George
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 3946902
zbMATH Keywords: Skorokhod space
zbMATH Keywords: tight sequence of processes
zbMATH Keywords: Lévy continuity theorem
zbMATH Keywords: nuclear space of distributions
zbMATH Keywords: asymptotic analysis of particle systems
MaRDI profile type: Publication
full work available at URL: 0022-1236(86)90078-9
OpenAlex ID: W2084568013
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description: scientific article; zbMATH DE number 3946902