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Freedman Inequality in Noncommutative Probability Spaces

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Abstract

The classical Freedman inequality, as a martingale extension of the Bernstein inequality, gives an upper bound for the tail probabilities of a supermartingale whose difference sequence is bounded above. In this paper, by employing a result of Lieb–Araki concerning the concavity of a certain map and construction of special projections corresponding to the event of the tail probabilities, we establish some Freedman inequalities for martingales in the setting of noncommutative probability spaces. As an application, among other things, we provide a noncommutative Bernstein-type inequality.

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Acknowledgements

The first author is supported by a grant from the Iran National Elites Foundation (INEF) for a postdoctoral fellowship under the supervision of the third author.

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Authors and Affiliations

  1. Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad, 91775, Iran

    A. Talebi & M. S. Moslehian

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

    Gh. Sadeghi

  3. Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran

    Gh. Sadeghi & M. S. Moslehian

Authors
  1. A. Talebi
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  2. Gh. Sadeghi
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  3. M. S. Moslehian
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Communicated by Palle Jorgensen.

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This article is part of the topical collection “Infinite-dimensional Analysis and Non-commutative Theory” edited by Marek Bozejko, Palle Jorgensen and Yuri Kondratiev.

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Talebi, A., Sadeghi, G. & Moslehian, M.S. Freedman Inequality in Noncommutative Probability Spaces. Complex Anal. Oper. Theory 16, 22 (2022). https://doi.org/10.1007/s11785-021-01186-4

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