Theoretical analysis demonstrates mean convergence for weighted sums of measurable operators, revealing generalized weak laws of large numbers for noncommutative and classical probability.
In this note, we extend and refine the sufficient part of the Feller weak law of large numbers by establishing mean convergence for weighted sums and the “maximum” of weighted sums of adapted sequences of measurable operators under regularly varying normalization. The corresponding convergence in measure results are obtained as immediate consequences of these stronger mean convergence results. The results are new even in the classical setting of random variables.
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Nguyen Van Quang (2026) studied this question.
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