  In this paper additive bi-free convolution is defined for general Borel
probability measures, and the limiting distributions for sums of bi-free pairs
of selfadjoint commuting random variables in an infinitesimal triangular array
are determined. These distributions are characterized by their bi-freely
infinite divisibility, and moreover, a transfer principle is established for
limit theorems in classical probability theory and Voiculescu's bi-free
probability theory. Complete descriptions of bi-free stability and fullness of
planar probability distributions are also set down. All these results reveal
one important feature about the theory of bi-free probability that it parallels
the classical theory perfectly well. The emphasis in the whole work is not on
the tool of bi-free combinatorics but only on the analytic machinery.
