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Mixing sequences, and mixingales in quantum probability spaces

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We introduce some measures of the dependence such as the strong mixing and uniform mixing coefficients in von Neumann algebras and then define the noncommutative strong and uniform mixing sequences. We establish some notable nonncommutative mixing inequalities such as Ibragimov inequality. Moreover, we extend the notion of mixingale sequence to the noncommutative content and demonstrate a noncommutative \(L_1\) and weak law of large numbers for uniformly integrable \(L_1\)-mixingale sequences. In addition, we investigate the noncommutative \(L_p\)-near-epoch dependence and provide some conditions under which a noncommtative \(L_p\)-near-epoch dependent sequence is a noncommutative \(L_p\)-mixingale. Finally, we introduce the concept of noncommutative \(L_p\)-approximability and show that in the setting of quantum (noncommutative) probability spaces, an \(L_r\)-bounded and \(L_0\)-approximable sequence is \(L_p\)-approximable for \(1\le p\) and \(2p< r<\infty \).

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Acknowledgements

The authors thank Professor Yong Jiao for his valuable comments. The authors would like to sincerely thank the referee for several comments and suggestions improving the paper

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Authors and Affiliations

  1. Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad, 91775, Iran

    Mohammad Sal Moslehian & Ghadir Sadeghi

  2. Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran

    Ghadir Sadeghi

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  1. Mohammad Sal Moslehian
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  2. Ghadir Sadeghi
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Correspondence to Ghadir Sadeghi.

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Moslehian, M.S., Sadeghi, G. Mixing sequences, and mixingales in quantum probability spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 89 (2022). https://doi.org/10.1007/s13398-022-01234-4

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  • DOI: https://doi.org/10.1007/s13398-022-01234-4

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