  We study two-faced families of random variables having bi-free infinitely
divisible distributions. We prove a limit theorem of the sums of bi-free
two-faced pairs of random variables within a triangular array. Then, by using
the full Fock space operator model, we show that a two-faced pair of random
variables has a bi-free (additive) infinitely divisible distribution if and
only if its distribution is the limit distribution in our limit theorem.
Finally, we characterize the bi-free (additive) infinite divisibility of the
distribution of a two-faced pair of random variables in terms of bi-free Levy
processes.
