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Reverse martingales in Riesz spaces. (English) Zbl 1198.60021

Grobler, J.J. (ed.) et al., Operator algebras, operator theory and applications. Proceedings of the 18th international workshop on operator theory and applications (IWOTA), Potchefstroom, South Africa, July 3–6, 2007. Basel: Birkhäuser (ISBN 978-3-0346-0173-3/hbk; 978-3-0346-0174-0/ebook). Operator Theory: Advances and Applications 195, 213-230 (2010).
The authors study reverse martingales in the measure-free setting of a Dedekind complete Riesz space \(E\) with a weak order unit (WOU), i.e., \(e\in E_+\) (the positive cone of \(E\)), and for all \(f\in E_+\), \(f\wedge ne\uparrow f\) \((n\in\mathbb{N})\). In Section 2 , reverse filtrations, reverse adapted sequences and martingales on Riesz spaces with a WOU are defined as well as the space of reverse adapted sequences and, its subspace, the space of reverse martingales. In Section 3, analogues of the Krickeberg and Bless decompositions are obtained for reverse martingales, in Section 4, a Riesz space upcrossing theorem for reverse martingales in a Riesz space with a WOU is given which, in turn, is used in Section 5 to derive reverse martingale convergence theorems in the Riesz space setting, in particular it is shown that bounded reverse martingales are convergent.
For the entire collection see [Zbl 1181.47005].

MSC:

60G48 Generalizations of martingales
46A40 Ordered topological linear spaces, vector lattices
47B60 Linear operators on ordered spaces
60G42 Martingales with discrete parameter
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