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On Azuma-Type Inequalities for Banach Space-Valued Martingales

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Abstract

In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space X is linearly isomorphic to a p-uniformly smooth space (\(1<p\le 2\)) if and only if an Azuma-type inequality holds for X-valued martingales. This can be viewed as a generalization of Pinelis’ work on an Azuma inequality for martingales with values in 2-uniformly smooth spaces. Secondly, an Azuma-type inequality for self-normalized sums will be presented. Finally, some further inequalities for Banach space-valued martingales, such as moment inequalities for double indexed dyadic martingales and De la Peña-type inequalities for conditionally symmetric martingales, will also be discussed.

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Acknowledgements

The author is partially supported by Natural Science Foundation of China (Grant No. 12071240). I would like to thank the Yau Mathematical Sciences Center of Tsinghua University for providing a great opportunity for me to complete this work. I am also grateful to the Associate Editor and the referees for their consideration.

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  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Sijie Luo

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  1. Sijie Luo
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Luo, S. On Azuma-Type Inequalities for Banach Space-Valued Martingales. J Theor Probab 35, 772–800 (2022). https://doi.org/10.1007/s10959-021-01086-5

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