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

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Publisher
Springer Journals
Copyright
Copyright © 2014 by 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

References

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