Distributional inequalities for noncommutative martingales

TL;DR

This paper develops distributional estimates for noncommutative martingales, extending classical inequalities into the operator setting and introducing new extrapolation theorems for broader applications.

Contribution

It introduces a novel operator-theoretic approach to establish distributional inequalities for noncommutative martingales, generalizing classical probabilistic results.

Findings

Distributional versions of key martingale inequalities established.

New extrapolation theorems for noncommutative settings introduced.

Applications include inequalities in symmetric quasi-Banach operator spaces.

Abstract

We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a method to build the noncommutative and classical probabilistic inequalities in an entirely operator theoretic way.