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

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Abstract

In a Dedekind complete Riesz space, E, we show that if \((P_n)\) is a sequence of band projections in E then

$$\begin{aligned} \limsup \limits _{n\rightarrow \infty } P_n - \liminf \limits _{n\rightarrow \infty } P_n = \limsup \limits _{n\rightarrow \infty } P_n(I-P_{n+1}). \end{aligned}$$

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.

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  1. School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, Johannesburg, South Africa

    Nyasha Mushambi, Bruce A. Watson & Bertin Zinsou

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  1. Nyasha Mushambi
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  2. Bruce A. Watson
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  3. Bertin Zinsou
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Correspondence to Bruce A. Watson.

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B. A. Watson: Supported in part by the Centre for Applicable Analysis and Number Theory and by NRF Grant No. IFR2011032400120.

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Mushambi, N., Watson, B.A. & Zinsou, B. Generalization of the theorems of Barndorff-Nielsen and Balakrishnan–Stepanov to Riesz spaces. Positivity 24, 753–760 (2020). https://doi.org/10.1007/s11117-019-00705-0

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  • DOI: https://doi.org/10.1007/s11117-019-00705-0

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