Ergodic theory and the strong law of large numbers on Riesz spaces
type: Item
instance of: scholarly article
title: Ergodic theory and the strong law of large numbers on Riesz spaces
zbMATH Open document ID: 1114.46015
author: Wen-Chi Kuo
author: Coenraad C. A. Labuschagne
author: Bruce A. Watson
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: The authors continue to develop Riesz space analogues of some central notions and results of probability theory. To give a flavour of the general set-up, note that the space \( L^1({\mathcal F}) \) of all integrable random variables on some probability space \( (\Omega,{\mathcal F},P) \) is not only a Banach space, but also a vector lattice (=: Riesz space) containing the constants, and that the conditional expectation \( E^{{\mathcal G}} : L^1({\mathcal F}) \to L^1({\mathcal F}) \) with respect to some sub-\(\sigma\)-algebra \( {\mathcal G} \subseteq {\mathcal F} \) is a linear operator which is not only norm continuous, but also positive, order continuous and idempotent with \( E^{{\mathcal G}}(L^1({\mathcal F})) = L^1({\mathcal G}) \) and that it preserves the constants. Dropping the probability space, the authors consider a general Riesz space \( E \) with a weak order unit and define a conditional expectation \( T : E \to E \) to be a positive order continuous projection for which \( T(E) \) is a Dedekind complete Riesz subspace of \( E \) and \( Te \) is a weak order unit whenever \( e \in E \) is a weak order unit. What is even more striking is the new notion of independence of Riesz subspaces with respect to a conditional expectation operator \( T : E \to E \), which corresponds perfectly to the notion of conditional independence of sub-\(\sigma\)-algebras with respect to a sub-\(\sigma\)-algebra \( {\mathcal G} \subseteq {\mathcal F} \). In this general setting, the authors prove abstract versions of ergodic theorems, of Kolmogorov's Zero-One Law, and of the Strong Law of Large Numbers for sequences of the form \( \frac{1}{n} \sum_{j=0}^{n-1} S^jf \) where \( f \in E \) and \( S : E \to E \) is an order continuous Riesz homomorphism.
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46N30
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 5078936
zbMATH Keywords: conditional expectation
zbMATH Keywords: ergodic theorem
zbMATH Keywords: law of large numbers
zbMATH Keywords: zero-one law
zbMATH Keywords: Riesz space
zbMATH Keywords: projections
reviewed by: Klaus D. Schmidt
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.01.056
OpenAlex ID: W1974350226
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DOI: 10.1016/J.JMAA.2006.01.056
description: scientific article; zbMATH DE number 5078936