Martingales in Banach Lattices |
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Authors: | Email author" target="_blank">Vladimir?G?TroitskyEmail author |
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Institution: | (1) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G2G1, Canada |
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Abstract: | In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive
projections (En) on a Banach lattice F is said to be a filtration if EnEm = En∧ m. A sequence (xn) in F is a martingale if Enxm = xn whenever n ≤ m. Denote by M = M(F, (En)) the Banach space of all norm uniformly bounded martingales. It is shown that if F doesn’t contain a copy of c0 or if every En is of finite rank then M is itself a Banach lattice. Convergence of martingales is investigated and a generalization of Doob Convergence Theorem is
established. It is proved that under certain conditions one has isometric embeddings
. Finally, it is shown that every martingale difference sequence is a monotone basic sequence.
Mathematics Subject Classification (2000). 60G48, 46B42 |
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Keywords: | Banach lattice filtration martingale |
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