Based on a maximal inequality-type result of Cuculescu, we establish some noncommutative maximal inequalities such as the Hajék–Penyi and Etemadi inequalities. In addition, we present a noncommutative Kolmogorov-type inequality by showing that if are successively independent self-adjoint random variables in a noncommutative probability space such that and , where , then, for any , there exists a projection such that
As a result, we investigate the relation between the convergence of a series of independent random variables and the corresponding series of their variances.
"Etemadi and Kolmogorov Inequalities in Noncommutative Probability Spaces." Michigan Math. J. 68 (1) 57 - 69, April 2019. https://doi.org/10.1307/mmj/1541667627