Abstract
We introduce the notion of acceptable noncommutative random variables and investigate their essential properties. More precisely, we provide several efficient estimation of tail probabilities of sums of noncommutative random variables under some mild conditions. Moreover, we investigate the complete convergence of a sequence of the form [Formula: see text] in which [Formula: see text] is a sequence of acceptable noncommutative random variables. Our results extend some classical inequalities as well.
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