2017/05/16 by Takahiro Hasebe, Hasebe, Takahiro, Hao-Wei Huang +3
Mathematics · #46L54 #60E07 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1705.05523
openalex publication_date 2017/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.