Large Deviation Inequalities for Noncommutative Martingales

TL;DR

This paper develops noncommutative large deviation inequalities for noncommutative martingales and independent variables, extending classical probabilistic results to the noncommutative setting.

Contribution

It introduces noncommutative analogs of large deviation inequalities, including a noncommutative Gordin's decomposition and ergodic theorem, for martingales and independent variables.

Findings

Characterized exponential integrability via large deviation inequalities.

Established deviation inequalities based on $L_{p}$-boundedness.

Derived a noncommutative ergodic theorem using deviation inequalities.

Abstract

We establish noncommutative analogs of some well-known large deviation inequalities for noncommutative random variables. Firstly, for the noncommutative independent case, we characterize the uniformly exponential integrability of random variables in terms of large deviation inequalities. Secondly, for noncommutative martingale differences, we establish two deviation inequalities according to the exponential integrability and $L_{p}$-boundedness of the martingale differences, respectively. Finally, we establish a noncommutative version of Gordin's decomposition, which enables us to derive a noncommutative ergodic theorem via deviation inequalities for noncommutative martingales.