  In this article, we introduce the notion of free subexponentiality, which
extends the notion of subexponentiality in the classical probability setup to
the noncommutative probability spaces under freeness. We show that
distributions with regularly varying tails belong to the class of free
subexponential distributions. This also shows that the partial sums of free
random elements having distributions with regularly varying tails are tail
equivalent to their maximum in the sense of Ben Arous and Voiculescu [Ann.
Probab. 34 (2006) 2037-2059]. The analysis is based on the asymptotic
relationship between the tail of the distribution and the real and the
imaginary parts of the remainder terms in Laurent series expansion of Cauchy
transform, as well as the relationship between the remainder terms in Laurent
series expansions of Cauchy and Voiculescu transforms, when the distribution
has regularly varying tails.
