Limit theorems in bi-free probability theory

• 成果类型: Article
• 署名作者: Hasebe, Takahiro; Huang, Hao-Wei; Wang, Jiun-Chau
• 署名单位: Hokkaido University; National Sun Yat Sen University; University of Saskatchewan
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0825-6
• 发表日期: 2018
• 页码: 1081-1119
• 关键词: analog

摘要

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 self-adjoint 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 are given and fullness of planar probability distributions is studied. 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.