Convergence in Riesz spaces with conditional expectation operators

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis 2015-09, Vol.19 (3), p.647-657
• Main Authors: Azouzi, Youssef, Kuo, Wen-Chi, Ramdane, Kawtar, Watson, Bruce A.
• Format: Article
• Language: English
• Subjects: Applied mathematics; Calculus of Variations and Optimal Control; Optimization; Econometrics; Fourier Analysis; Mathematics; Mathematics and Statistics; Operator Theory; Potential Theory; Random variables; Studies; Vector space
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-014-0320-6

A conditional expectation, 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 ) a Dedekind complete Riesz subspace of E . The concepts of strong convergence and convergence in probability are extended to this setting as T -strongly convergence and convergence in T -conditional probability. Critical to the relating of these types of convergence are the concepts of uniform integrability and norm boundedness, generalized as T -uniformity and T -boundedness. Here we show that if a net is T -uniform and convergent in T -conditional probability then it is T -strongly convergent, and if a net is T -strongly convergent then it is convergent in T -conditional probability. For sequences we have the equivalence that a sequence is T -uniform and convergent in T -conditional probability if and only if it is 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.