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 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.
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Acknowledgements
The authors would like to thank anonymous reviewers for providing comments and suggestions which improve the quality of this paper. The first-named author is supported by JSPS Grant-in-Aid for Young Scientists (B) 15K17549. The second-named author was supported through a grant from the Ministry of Science and Technology in Taiwan MOST-104-2115-M-110-011-MY2 and a faculty startup grant from the National Sun Yat-sen University. The third-named author was supported by the NSERC Canada Discovery Grant RGPIN-2016-03796.
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Hasebe, T., Huang, HW. & Wang, JC. Limit theorems in bi-free probability theory. Probab. Theory Relat. Fields 172, 1081–1119 (2018). https://doi.org/10.1007/s00440-017-0825-6
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DOI: https://doi.org/10.1007/s00440-017-0825-6
Keywords
- Bi-free limit theorem
- Bi-free infinitely divisible distributions
- Bi-freely stable distributions
- Full distributions
Mathematics Subject Classification
- Primary 46L54
- Secondary 60E07