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Remarks on a Free Analogue of the Beta Prime Distribution

  • Hiroaki YoshidaEmail author
Article
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Abstract

We introduce the free analogue of the classical beta prime distribution by the multiplicative free convolution of the free Poisson and the reciprocal of free Poisson distributions, and related free analogues of the classical F, T, and beta distributions. We show the rationales of our free analogues via the score functions and the potentials. We calculate the moments of the free beta prime distribution explicitly in combinatorial fashion by using non-crossing linked partitions and demonstrate that the free beta prime distribution belongs to the class of the free negative binomials in the free Meixner family.

Keywords

Free beta prime distribution Free F-distribution Free T-distribution Free beta distribution Non-crossing linked partitions Free Meixner family 

Mathematics Subject Classification (2010)

46L54 60E05 

Notes

Acknowledgements

The author is very grateful to an anonymous referee who has read carefully and checked formulae and pointed out inaccurate parts in the original submission. The author also thanks T. Hasebe and N. Sakuma for their helpful discussions. This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) JP26400112.

References

  1. 1.
    Anshelevich, M.: Free martingale polynomials. J. Funct. Anal. 201(1), 228–261 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bai, Z., Silverstein, J.W.: Spectral Analysis of Large Dimensional Random Matrices. Springer Series in Statistics, 2nd edn. Springer, New York (2010)zbMATHGoogle Scholar
  3. 3.
    Banica, T., Belinschi, S.T., Capitaine, M., Collins, B.: Free Bessel Laws. Can. J. Math. 63(1), 3–37 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bercovici, H., Pata, V.: Stable laws and domains of attraction in free probability theory (with an appendix by Philippe Biane). Ann. Math. 149(2), 1023–1060 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Biane, P.: Logarithmic Sobolev inequalities, matrix models and free entropy. Acta Math. Sin. 19(3), 497–506 (2003)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Biane, P., Speicher, R.: Free diffusions, free energy and free Fisher information. Ann. Inst. H. Poincaré Probab. Stat. 37(6), 581–606 (2001)zbMATHGoogle Scholar
  7. 7.
    Blower, G.: Random Matrices: High Dimensional Phenomena, London Mathematical Society. Lecture Notes Series, vol. 367. Cambridge University Press, London (2009)Google Scholar
  8. 8.
    Bożejko, M., Bryc, W.: On a class of free Levy laws related to a regression problem. J. Funct. Anal. 236(1), 59–77 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bożejko, M., Lytvynov, E.: Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization. Commun. Math. Phys. 292(1), 99–129 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bryc, W.: Markov processes with free Meixner laws. Stoch. Process. Appl. 120(8), 1393–1403 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chen, W., Wu, S., Yan, C.: Linked partitions and linked cycles. Eur. J. Combin. 29(6), 1408–1426 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dykema, K.: Multilinear function series and transforms in free probability theory. Adv. Math. 208(1), 351–407 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ehrenborg, R., Readdy, M.: Juggling and applications to \(q\)-analogues. Discrete Math. 157(1–3), 107–125 (1996)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ejsmont, W.: Laha–Lukacs properties of some free processes. Electron. Commun. Probab. 17(13), 1–8 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ejsmont, W.: Characterizations of some free random variables by properties of conditional moments of third degree polynomials. J. Theor. Probab. 27(3), 915–931 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ejsmont, W., Franz, U., Szpojankowski, K.: Convolution, subordination and characterization problems in noncommutative probability. Indiana Univ. Math. J. 66(1), 237–257 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ejsmont, W., Lehner, F.: Sample variance in free probability. J. Funct. Anal. 273(7), 2488–2520 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Flajolet, P.: Combinatorial aspects of continued fractions. Discrete Math. 32(2), 125–161 (1980)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Haagerup, U.: On Voiculescu’s \(R\)- and \(S\)-transforms for free non-commuting random variables. In: Voiculescu, D. (ed.) Free Probability Theory (Waterloo, ON, 1995) Fields Institute Communications, vol. 12, pp. 127–148. American Mathematical Society, Providence (1997)Google Scholar
  20. 20.
    Haagerup, U., Schultz, H.: Brown measures of unbounded operators affiliated with a finite von Neumann algebra. Math. Scand. 100(2), 209–263 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hasebe, T., Szpojankowski, K.: On free generalized inverse Gaussian distributions. Complex Anal. Oper. Theory (2018).  https://doi.org/10.1007/s11785-018-0790-9 Google Scholar
  22. 22.
    Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs, vol. 77. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  23. 23.
    Hiwatashi, O., Kuroda, T., Nagsa, M., Yoshida, H.: The free analogue of noncentral chi-square distributions and symmetric quadratic forms in free random variables. Math. Z. 230(1), 63–77 (1999)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kreweras, G.: Sur les partitions non croisées d’un cycle. Discrete Math. 1(4), 333–350 (1972)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lukacs, E.: A characterization of the gamma distribution. Ann. Math. Stat. 26(2), 319–324 (1955)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Mingo, J., Speicher, R.: Free Probability and Random Matrices. Springer, New York (2017)zbMATHGoogle Scholar
  27. 27.
    Nemoto, A., Yoshida, H.: The free logarithmic Sobolev and the free transportation cost inequalities by time integrations. Infin. Dimen. Anal. Quantum Probab. Relat. Top. 17(3), 1450022 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Nica, A.: \(R\)-transforms of free joint distributions and non-crossing partitions. J. Funct. Anal. 135(2), 271–296 (1996)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Nica, A.: Non-crossing linked partitions, the partial order \(\ll \) on \(NC(n)\) and the \(S\)-transform. Proc. Am. Math. Soc. 138(4), 1273–1285 (2010)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability London Mathematical Society Lecture Note Series, vol. 335. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  31. 31.
    Saitoh, N., Yoshida, H.: The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory. Probab. Math. Stat. 21(1), 159–170 (2001)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Simion, R.: Noncrossing partitions. Discrete Math. 217(1–3), 367–409 (2000)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Speicher, R.: Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298(1), 611–628 (1994)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Szpojankowski, K.: On the Lukacs property for free random variables. Stud. Math. 228(1), 55–7 (2015)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Szpojankowski, K.: A constant regression characterization of the Marchenko–Pastur law. Probab. Math. Stat. 36(1), 137–145 (2016)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Szpojankowski, K.: On the Matsumoto–Yor property in free probability. J. Math. Anal. Appl. 445(1), 374–393 (2017)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Szpojankowski, K., Wesolowski, J.: Dual Lukacs regressions for non-commutative variables. J. Funct. Anal. 266(1), 36–54 (2014)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Voiculescu, D.: Symmetries of some reduced free product \(C^*\)-algebras. In: Araki, H., Moore, C., Stratila, S., Voiculescu, D. (eds.) Operator Algebras and their Connections with Topology and Ergodic Theory Lecture Notes in Mathematics 1132. Springer, Berlin (1985)Google Scholar
  39. 39.
    Voiculescu, D.: Addition of certain noncommuting random variables. J. Funct. Anal. 66(3), 323–346 (1986)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Voiculescu, D.: Multiplication of certain non-commuting random variables. J. Oper. Theory 18(2), 2230–235 (1987)zbMATHGoogle Scholar
  41. 41.
    Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory I. Commun. Math. Phys. 155(1), 71–92 (1993)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Voiculescu, D.: Free entropy. Bull. Lond. Math. Soc. 34(3), 257–278 (2002)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Voiculescu, D., Dykema, K., Nica, A.: Free Random Variables. A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups. CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992)zbMATHGoogle Scholar
  44. 44.
    Yano, F., Yoshida, H.: Some set partition statistics in non-crossing partitions and generating functions. Discrete Math. 307(24), 3147–3160 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Information SciencesOchanomizu UniversityBunkyoJapan

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