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Positivity

, Volume 9, Issue 3, pp 437–456 | Cite as

Martingales in Banach Lattices

Dedicated to the memory of Yuri Abramovich, my friend and advisor
  • Vladimir G. TroitskyEmail author
Article

Abstract

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (E n ) on a Banach lattice F is said to be a filtration if E n E m = Enm. A sequence (x n ) in F is a martingale if E n x m = x n whenever nm. Denote by M = M(F, (E n )) 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 E n 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 \(F \hookrightarrow M \hookrightarrow F^{**}\). Finally, it is shown that every martingale difference sequence is a monotone basic sequence.

Keywords

Banach lattice filtration martingale 

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© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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