In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.

We give a description of infinitely divisible compactly supported probability measures relative to the multiplicative free convolutions on the positive half-line and on the unit circle. A new proof is provided for the analogous result for additive free convolution.

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We show that the subordination results of D. Voiculescu and Ph. Biane can be deduced from a continuity property of fixed points for analytic functions.