# Convergence in Riesz spaces with conditional expectation operators

Convergence in Riesz spaces with conditional expectation operators A conditional expectation, \$\$T\$\$ T , on a Dedekind complete Riesz space with weak order unit is a positive order continuous projection which maps weak order units to weak order units and has \$\$R(T)\$\$ R ( T ) a Dedekind complete Riesz subspace of \$\$E\$\$ E . The concepts of strong convergence and convergence in probability are extended to this setting as \$\$T\$\$ T -strongly convergence and convergence in \$\$T\$\$ T -conditional probability. Critical to the relating of these types of convergence are the concepts of uniform integrability and norm boundedness, generalized as \$\$T\$\$ T -uniformity and \$\$T\$\$ T -boundedness. Here we show that if a net is \$\$T\$\$ T -uniform and convergent in \$\$T\$\$ T -conditional probability then it is \$\$T\$\$ T -strongly convergent, and if a net is \$\$T\$\$ T -strongly convergent then it is convergent in \$\$T\$\$ T -conditional probability. For sequences we have the equivalence that a sequence is \$\$T\$\$ T -uniform and convergent in \$\$T\$\$ T -conditional probability if and only if it is \$\$T\$\$ T -strongly convergent. These results are applied to Riesz space martingales and are applicable to stochastic processes having random variables with ill-defined or infinite expectation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Convergence in Riesz spaces with conditional expectation operators

, Volume 19 (3) – Dec 23, 2014
11 pages

/lp/springer_journal/convergence-in-riesz-spaces-with-conditional-expectation-operators-VXb7DOSYi6
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0320-6
Publisher site
See Article on Publisher Site

### Abstract

A conditional expectation, \$\$T\$\$ T , on a Dedekind complete Riesz space with weak order unit is a positive order continuous projection which maps weak order units to weak order units and has \$\$R(T)\$\$ R ( T ) a Dedekind complete Riesz subspace of \$\$E\$\$ E . The concepts of strong convergence and convergence in probability are extended to this setting as \$\$T\$\$ T -strongly convergence and convergence in \$\$T\$\$ T -conditional probability. Critical to the relating of these types of convergence are the concepts of uniform integrability and norm boundedness, generalized as \$\$T\$\$ T -uniformity and \$\$T\$\$ T -boundedness. Here we show that if a net is \$\$T\$\$ T -uniform and convergent in \$\$T\$\$ T -conditional probability then it is \$\$T\$\$ T -strongly convergent, and if a net is \$\$T\$\$ T -strongly convergent then it is convergent in \$\$T\$\$ T -conditional probability. For sequences we have the equivalence that a sequence is \$\$T\$\$ T -uniform and convergent in \$\$T\$\$ T -conditional probability if and only if it is \$\$T\$\$ T -strongly convergent. These results are applied to Riesz space martingales and are applicable to stochastic processes having random variables with ill-defined or infinite expectation.

### Journal

PositivitySpringer Journals

Published: Dec 23, 2014

## 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
discover and read the research
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 lists to

Export lists, citations

Abstract access only

18 million full-text articles

Print

20 pages / month

PDF Discount

20% off