We show that the three series theorem of classical probability holds for free random variables as well. Sufficiency of the three series condition was known under more general circumstances, so only necessity is at issue.
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Bercovici, H. Series of Free Random Variables. J Theor Probab 18, 957–965 (2005). https://doi.org/10.1007/s10959-005-7540-4
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DOI: https://doi.org/10.1007/s10959-005-7540-4