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arXiv · 1705.05523

Limit theorems in bi-free probability theory

Takahiro Hasebe·Hao-Wei Huang·Jiun-Chau Wang
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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.

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Takahiro Hasebe, Hao-Wei Huang, Jiun-Chau Wang. 2017-05-16. Limit theorems in bi-free probability theory. https://arxiv.org/abs/1705.05523

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