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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