Bernoulli Processes in Riesz spaces

Abstract

The action and averaging properties of conditional expectation operators are studied in the, measure-free, Riesz space, setting of Kuo, Labuschagne and Watson [{Conditional expectations on Riesz spaces}, J. Math. Anal. Appl., 303 (2005), 509-521] but on the abstract $L^2$ space, ${\mathcal{L}}^2(T)$ introduced by Labuschagne and Watson [{ Discrete Stochastic Integration in Riesz Spaces}, Positivity, 14, (2010), 859 - 575]. In this setting it is shown that conditional expectation operators leave ${\mathcal{L}}^2(T)$ invariant and the Bienaymé equality and Tchebichev inequality are proved. From this foundation Bernoulli processes are considered. Bernoulli's strong law of large numbers and Poisson's theorem are formulated and proved.

Ordered Structures and Applications: Positivity VII. Trends in Mathematics 263-274, 2016