Limit theorems in bi-free probability theory

TL;DR

This paper develops limit theorems in bi-free probability, establishing distributions for sums of bi-free pairs, and shows how classical and bi-free theories are closely aligned, emphasizing analytic methods over combinatorics.

Contribution

It introduces additive bi-free convolution for general measures, characterizes bi-freely infinitely divisible distributions, and establishes a transfer principle linking classical and bi-free limit theorems.

Findings

Characterization of limiting distributions via bi-free infinite divisibility

Establishment of a transfer principle between classical and bi-free probability

Descriptions of bi-free stability and fullness of planar distributions

Abstract

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.