Lattices of random sets and progressivity
type: Item
instance of: scholarly article
title: Lattices of random sets and progressivity
zbMATH Open document ID: 0842.60013
DOI: 10.1016/0167-7152(94)00054-C
author: B. Gail Ivanoff
author: Ely Merzbach
author: Ioana Schiopu Kratina
published in: Statistics \& Probability Letters
publication date: 1995-02-22T00:00:00Z
review text: Let \(T\) be a complete separable metric space and let \({\mathcal B}\) be the Borel-\(\sigma\)-algebra on \(T\). Let \({\mathcal A}\) be a lattice of closed subsets of \(T\) which is closed under countable intersections and which satisfies a separability assumption. Let \((\Omega,{\mathcal F})\) be a measurable space. The authors define the notions of (simple) random set, of (simple) stopping set with values in \({\mathcal A}\) and of the \(\sigma\)-algebra of progressively measurable sets in \({\mathcal F} \otimes {\mathcal B}\) (the latter two notions with respect to a family of sub-\(\sigma\)-algebras of \({\mathcal F}\) indexed by the elements of \({\mathcal A}\)). The definition of stopping sets given by the authors generalizes the notion of co-compatible sets given by \textit{Yu. A. Rozanov} [``Markov random fields'' (1982; Zbl 0498.60057)], but is not the same as the definition given by \textit{T. G. Kurtz} [Ann. Probab. 8, 675-681 (1980; Zbl 0442.60045)]. The authors show that a random set is a stopping set iff the associated set in the product space \({\mathcal F} \otimes {\mathcal B}\) is progressively measurable.
reviewed by: M.Dozzi
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 724076
zbMATH Keywords: set-indexed processes
zbMATH Keywords: random set
zbMATH Keywords: stopping set
zbMATH Keywords: sigma-algebra of progressively measurable sets
MaRDI profile type: Publication
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full work available at URL: 0167-7152(94)00054-c
OpenAlex ID: W1987807343
description: scientific article; zbMATH DE number 724076