Noncommutative Fuk–Nagaev Inequalities and Their Applications

Abstract

We establish a noncommutative extension of the Fuk–Nagaev inequalities for random variables in a noncommutative probability space. As applications, we first obtain noncommutative versions of Bennett inequality and Rosenthal inequality, and secondly, we study the weak law of large numbers for weighted sums in a noncommutative probability space and the weak law of large numbers for weighted sums of probability measures corresponding to the free additive convolutions.