Noncommutative martingale concentration inequalities

TL;DR

This paper develops noncommutative martingale concentration inequalities, extending classical results to the noncommutative setting with applications to various inequalities involving martingales and supermartingales.

Contribution

It introduces new noncommutative Azuma, Hoeffding, McDiarmid, and Bernstein inequalities, broadening the scope of concentration results in noncommutative probability theory.

Findings

Established a noncommutative Azuma inequality under Lipschitz condition.

Derived noncommutative Hoeffding and McDiarmid inequalities.

Provided a noncommutative Bernstein inequality and $L_p$-norm bounds.

Abstract

We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type inequality. We also provide a noncommutative Azuma inequality for noncommutative supermartingales in which instead of a fixed upper bound for the variance we assume that the variance is bounded above by a linear function of variables. We then employ it to deduce a noncommutative Bernstein inequality and an inequality involving $L_p$-norm of the sum of a martingale difference.