Optimum bounds for the distributions of martingales in Banach spaces

TL;DR

This paper extends exponential inequalities for sums of independent variables to martingales in 2-smooth Banach spaces, providing optimal bounds on moments and sums in infinite-dimensional settings, with novel results even for real-valued cases.

Contribution

It introduces a general method to derive optimal moment bounds for martingales in 2-smooth Banach spaces, extending classical inequalities to infinite-dimensional contexts.

Findings

Derived optimal bounds for martingale moments in 2-smooth Banach spaces.

Extended inequalities to sums of independent vectors in separable Banach spaces.

Provided new bounds for real-valued martingales and supermartingales.

Abstract

A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in the 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of the sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on the infinite-dimensional martingales, most of the results seem to be new even for the one-dimensional ones. Moreover, the bounds on the Rosenthal-Burkholder type of moments seem to be to certain extent new even for the sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.