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We study the tail behavior of the law of the sum of free identically
distributed random variables affiliated to some W*-probability space with
its law having regularly varying tail. We introduce the notion of free
subexponentiality by extending the concept from the classical setup and
show that the laws with regularly varying tail are subexponential. The
analysis is based on the relation between the remainder terms in Taylor
series expansions of Cauchy and Voiculescu transforms, when the law admits
moments. We study the tail behavior of the law of the sum of free
identically distributed random till certain finite order. We use this
analysis to extend Embrechts-Goldie-Veraverbeke theorem regarding the tail
equivalence of a free infinitely divisible law and its LC)vy measure in
the regular variation setup.
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