# Martingales in Banach Lattices

Martingales in Banach Lattices 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 = E n∧ m . A sequence (x n ) in F is a martingale if E n x m = x n whenever n ≤ m. 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 c 0 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Martingales in Banach Lattices

20 pages

/lp/springer-journals/martingales-in-banach-lattices-Ac4jftcg5k
Publisher site
See Article on Publisher Site

### 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 = E n∧ m . A sequence (x n ) in F is a martingale if E n x m = x n whenever n ≤ m. 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 c 0 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.

### Journal

PositivitySpringer Journals

Published: Feb 23, 2004

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just \$49/month

### Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

### Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

### Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

### Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
PubMed

Create folders to

Export folders, citations