Bernoulli Processes in Riesz spaces

2017/07/04 by Wen-Chi Kuo, Kuo, Wen-Chi, Jessica Joy Vardy +3
Mathematics · #47B60 #47B80 #60B12 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:47B60 #msc:47B80 #msc:60B12

paper · pdf · doi:10.48550/arxiv.1707.00968

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

arxiv created 2017/07/04 · arxiv updated 2017/07/05

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 L² space, \cal L²(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 \cal L²(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.

Related

• Strong completeness of a class of L2-type Riesz spaces
• Mixing inequalities in Riesz spaces
• Strong sequential completeness of the natural domain of a conditional expectation operator in Riesz spaces
• A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces
• The Kac formula and Poincaré recurrence theorem in Riesz spaces