Large Deviation Inequalities for Noncommutative Martingales

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.