Analytic aspects of the bi-free partial R-transform

It is shown that the bi-freely infinitely divisible laws, and only these laws, can be used to approximate the distributions of sums of identically distributed bi-free pairs of commuting faces. Furthermore, the necessary and sufficient conditions for this approximation are found. Bi-free convolution semigroups of measures and their Lévy–Khintchine representations are also studied here from an infinitesimal point of view. The proofs depend on the harmonic analysis machinery we developed for integral transforms of two variables, without reference to the combinatorics of moments and bi-free cumulants.