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.
Positivity – Springer Journals
Published: Dec 23, 2014
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