Unconditional martingale difference sequences in Banach spaces
type: Item
instance of: scholarly article
title: Unconditional martingale difference sequences in Banach spaces
zbMATH Open document ID: 1139.46012
author: Stuart F. Cullender
author: Coenraad C. A. Labuschagne
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-14T00:00:00Z
review text: A result of \textit{D. J. Aldous} [Math. Proc. Camb. Philos. Soc. 85, 117--123 (1979; Zbl 0389.46027)] says that \(L^p(Y)\) fails to have an unconditional basis if \(Y\) is not superreflexive. This paper attempts to provide weaker positive results in this direction. The starting point is an abstract version of the notion of a martingale in a Banach space \(X\) in which some uniformly bounded increasing sequence of projections on \(X\) plays the rôle of the conditional expectations in classical martingale theory in \(L^p\). Along with this notion comes an abstract version of a martingale difference sequence and of an unconditional martingale difference sequence. The authors show that an abstract martingale difference sequence \((d_i)\) is unconditional if and only if the associated coordinate space \(\{(\alpha_i): \sum_i \alpha_i d_i\) converges\} is an order continuous Banach lattice. As the authors point out in a corrigendum to this paper, there is a serious mathematical error on page 1300 that adversely affects the inequality stated in the abstract, Theorems 5.2 and 7.2 and Corollaries 7.3 and 7.4. They also point out that the affected results still hold for disjoint sequences and that corrected results for general sequences can be found in the first-named author's Ph.D. thesis.
reviewed by: Dirk Werner
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 5125972
zbMATH Keywords: abstract martingale difference sequence
zbMATH Keywords: unconditional basis
zbMATH Keywords: tensor product of Banach lattices
zbMATH Keywords: Bochner \(L^p\)-space
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.03.061
OpenAlex ID: W2003641466
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DOI: 10.1016/J.JMAA.2006.03.061
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description: scientific article; zbMATH DE number 5125972