Quantitative laws of large numbers in noncommutative probability

Abstract In this article, we investigate the quantitative law of large numbers for noncommutative random variables. Firstly, we establish a Baum–Katz theorem for noncommutative successively independent random variables, which resolves an open problem posed by Stoica [48]. Our approach differs from the classical treatment but relies on the theory of asymmetric maximal inequality for noncommutative martingales. Additionally, we derive a moderate deviation inequality for noncommutative successively independent sequences, and extend this result together with the Baum–Katz theorem to noncommutative martingales. Finally, we conclude the article by applying our results to derive a noncommutative Marcinkiewicz–Zygmund-type strong laws of large numbers theorem, which extends the result of Łuczak [37] in some aspects.