Best possible bounds of the von Bahr--Esseen type

TL;DR

This paper extends the von Bahr--Esseen bound to a broader class of moment functions with optimal constants, providing new insights into martingale moments and applications in measure concentration and normed spaces.

Contribution

It introduces the best possible bounds for a wide class of moment functions, generalizing the von Bahr--Esseen inequality with optimal constants.

Findings

Extended von Bahr--Esseen bounds to general moment functions

Derived measure concentration inequalities for Lipschitz functions

Discussed relations with smooth and convex normed spaces

Abstract

The well-known von Bahr--Esseen bound on the absolute $p$th moments of martingales with $p\in(1,2]$ is extended to a large class of moment functions, and now with a best possible constant factor (which depends on the moment function). This result appears to be new even for the power moments. As an application, measure concentration inequalities for separately Lipschitz functions on product spaces are obtained. Relations with $p$-uniformly smooth and $q$-uniformly convex normed spaces are discussed.