Best possible bounds of the von Bahr--Esseen type

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). 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.