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Limit theorems in bi-free probability theory

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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References

  1. Barczy, M., Pap, G.: Portmanteau theorem for unbounded measures. Stat. Probab. Lett. 76, 1831–1835 (2006)

    Article  MathSciNet  Google Scholar 

  2. Belinschi, S.T., Bercovici, H., Gu, Y., Skoufranis, P.: Analytic subordination for bi-free convolution. ArXiv: 1702.01673v1

  3. Bercovici, H., Pata, V.: Stable laws and domains of attraction in free probability theory. Ann. Math. 149, 1023–1060 (1999)

    Article  MathSciNet  Google Scholar 

  4. Bercovici, H., Pata, V.: A free analogue of Hinčin characterization of infinite divisibility. Proc. Am. Math. Soc. 128(4), 1011–1015 (2000)

    Article  Google Scholar 

  5. Bercovici, H., Voiculescu, D.: Free convolutions of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733–773 (1993)

    Article  MathSciNet  Google Scholar 

  6. Bercovici, H., Wang, J.-C.: The asymptotic behavior of free additive convolution. Oper. Matrices 2(1), 115–124 (2008)

    Article  MathSciNet  Google Scholar 

  7. Charlesworth, I., Nelson, B., Skoufranis, P.: On two-faced families of non-commutative random variables. Can. J. Math. 67(6), 1290–1325 (2015)

    Article  MathSciNet  Google Scholar 

  8. Charlesworth, I., Nelson, B., Skoufranis, P.: Combinatorics of bi-freeness with amalgamation. Commun. Math. Phys. 338(2), 801–847 (2015)

    Article  MathSciNet  Google Scholar 

  9. Chistyakov, G.P., Götze, F.: Limit theorems in free probability theory. I. Ann. Probab. 36(1), 54–90 (2008)

    Article  MathSciNet  Google Scholar 

  10. Feldheim, E.: Étude de la stabilité des lois de probabilité, Ph.D. Thesis, Faculté des Sciences de Paris. Paris, France (1937)

  11. Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley Publishing Company, Inc., Boston (1954)

    MATH  Google Scholar 

  12. Gu, Y., Huang, H.-W., Mingo, J.A.: An analogue of the Lévy-Hinčin formula for bi-free infinitely divisible distributions. Indiana Univ. Math. J. 65(5), 1795–1831 (2016)

    Article  MathSciNet  Google Scholar 

  13. Gu, Y., Skoufranis, P.: Bi-Boolean independence for pairs of algebras. ArXiv:1703.03072

  14. Huang, H.-W., Wang, J.-C.: Analytic aspects of bi-free partial \(R\)-transform. J. Funct. Anal. 271(4), 922–957 (2016)

    Article  MathSciNet  Google Scholar 

  15. Lévy, P.: Théorie de laddition des variables aléatoires, Monographies des probabilités, vol. 1, 2nd edn. Gauthier-Villars, Paris (1954)

    Google Scholar 

  16. Meerschaert, M.M., Scheffler, H.-P.: Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice, Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (2001). http://www.stt.msu.edu/~mcubed/RVbook.html

  17. Rvačeva, E. L.: On domains of attraction of multi-dimensional distributions, 1962 Select. Transl. Math. Statist. and Probability2 pp. 183–205 American Mathematical Society, Providence, RI

  18. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  19. Skoufranis, P.: Independences and partial \(R\)-transforms in bi-free probability. Ann. Inst. Henri Poincaré Probab. Stat. 52(3), 1437–1473 (2016)

    Article  MathSciNet  Google Scholar 

  20. Skoufranis, P.: A combinatorial approach to Voiculescu’s bi-free partial transforms. Pac. J. Math. 283(2), 419–447 (2016)

    Article  MathSciNet  Google Scholar 

  21. Voiculescu, D.V.: Free probability for pairs of faces I. Commun. Math. Phys. 332, 955–980 (2014)

    Article  MathSciNet  Google Scholar 

  22. Voiculescu, D.V.: Free probability for pairs of faces II: 2-variables bi-free partial \(R\)-transform and systems with rank \(\le \)1 commutation. Ann. Inst. Henri Poincaré Probab. Stat. 52(1), 1–15 (2016)

    Article  MathSciNet  Google Scholar 

  23. Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables. CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992)

    MATH  Google Scholar 

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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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Correspondence to Hao-Wei Huang.

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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