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Inequalities for martingales with respect to positive module operators

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Abstract

We study two inequalities which are concerned with \({\mathbf {T}}\)-martingales. More precisely, we establish the noncommutative Azuma inequality for \({\mathbf {T}}\)-martingales and noncommutative \({\mathbf {T}}\)-martingale deviation inequality using the Golden–Thompson inequality for a positive module operator. The inequalities in this paper are considered as generalizations of the inequalities for a conditional expectation and a state. As an application, we study some convergence results which are concerned with the weak law of large numbers for weighted sums in a noncommutative probability space.

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Acknowledgements

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2020R1C1C1A01003305). The author thanks the anonymous referees for valuable comments on this paper.

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  1. Department of Mathematics Education, Jeju National University, Jeju, 63243, South Korea

    Byoung Jin Choi

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  1. Byoung Jin Choi
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Correspondence to Byoung Jin Choi.

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Communicated by M. S. Moslehian.

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Choi, B.J. Inequalities for martingales with respect to positive module operators. Adv. Oper. Theory 5, 1455–1467 (2020). https://doi.org/10.1007/s43036-020-00054-w

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