On Azuma-type inequalities for Banach space-valued martingales

TL;DR

This paper establishes a characterization of Banach spaces via Azuma-type inequalities for martingales, extending existing results and introducing new inequalities for self-normalized sums and other martingale types.

Contribution

It provides a characterization of Banach spaces that admit Azuma-type inequalities, generalizes Pinelis' results, and introduces new inequalities for various classes of Banach space-valued martingales.

Findings

Banach space is linearly isomorphic to a p-uniformly smooth space if and only if Azuma-type inequality holds.

Presented Azuma-type inequality for self-normalized sums.

Derived moment inequalities for double indexed dyadic martingales and De la Peña-type inequalities for conditionally symmetric martingales.

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 ($1<p\leq 2$) if and only if an Azuma-type inequality holds for $X$-valued martingales. This can be viewed as a generalization of Pinelis' work on Azuma inequality for martingales with values in $2$-uniformly smooth space. Secondly, 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 the De la Pe\~{n}a-type inequalities for conditionally symmetric martingales, will also be discussed.