Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions

Abstract

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. 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.