| Abstract: |
| In free probability the notion of free convolution of probability distributions on $\mathbb{R}$ has played an important role since its inception by D. Voiculescu some $30$ years ago. In 2013, Voiculescu generalized the notion of free independence to study left and right actions on reduced free product spaces simultaneously, known as bi-free independence. One generalization of the free convolution to the bi-free setting is the bi-free convolution of planar probability distributions. In this talk, we will explain 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. We will also talk about bi-free L\`{e}vy-Khintchine representations from an infinitesimal point of view. The proofs depend on the bi-free harmonic analysis machinery that we developed for integral
transforms of two variables. If time permits, some recent developments in this direction will also be discussed. |
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