Generalization of the theorems of Barndorff-Nielsen and Balakrishnan-Stepanov to Riesz spaces

TL;DR

This paper extends classical theorems of probability to the setting of Dedekind complete Riesz spaces, providing new identities and conditional versions of the Borel-Cantelli theorem.

Contribution

It generalizes the theorems of Barndorff-Nielsen and Balakrishnan-Stepanov to Riesz spaces, introducing new identities and conditional extensions.

Findings

Established a new identity relating limsup and liminf of band projections.

Derived conditional versions of the First Borel-Cantelli Theorem in Riesz spaces.

Extended classical probabilistic theorems to a more general ordered space setting.

Abstract

In a Dedekind complete Riesz space, $E$, we show that if $(P_n)$ is a sequence of band projections in $E$ then $$\limsup\limits_{n\to \infty} P_n - \liminf\limits_{n\to \infty} P_n = \limsup\limits_{n\to \infty} P_n(I-P_{n+1}).$$ This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan-Stepanov generalizations of the First Borel-Cantelli Theorem.