Noncommutative characterization of free Meixner processes

Wiktor Ejsmont (Wroclaw University of Economics and University of Wroclaw)


In this article we give a purely noncommutative criterion for the characterization of free Meixner random variables. We prove that some families of free Meixner distributions can be described in terms of the conditional expectation, which has no classical analogue. We also show a generalization of Speicher's formula (relating moments and free cumulants) and establish a new relation in free probability.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-12

Publication Date: March 24, 2013

DOI: 10.1214/ECP.v18-2356


  • Akhiezer N.I., Glazman I.M.: Theory of Linear Operators in Hilbert Space. Ungar, New York, (1963).
  • Anshelevich, Michael. Free martingale polynomials. J. Funct. Anal. 201 (2003), no. 1, 228--261. MR1986160
  • Anshelevich, Michael. Orthogonal polynomials with a resolvent-type generating function. Trans. Amer. Math. Soc. 360 (2008), no. 8, 4125--4143. MR2395166
  • Anshelevich, Michael. Bochner-Pearson-type characterization of the free Meixner class. Adv. in Appl. Math. 46 (2011), no. 1-4, 25--45. MR2794011
  • Anshelevich, Michael; Młotkowski, Wojciech. Semigroups of distributions with linear Jacobi parameters. J. Theoret. Probab. 25 (2012), no. 4, 1173--1206. MR2993018
  • Barndorff-Nielsen, Ole E.; Thorbjørnsen, Steen. Self-decomposability and Lévy processes in free probability. Bernoulli 8 (2002), no. 3, 323--366. MR1913111
  • Benaych-Georges, Florent. Taylor expansions of $R$-transforms: application to supports and moments. Indiana Univ. Math. J. 55 (2006), no. 2, 465--481. MR2225442
  • Biane, Philippe. Processes with free increments. Math. Z. 227 (1998), no. 1, 143--174. MR1605393
  • Bożejko, Marek; Bryc, Włodzimierz. On a class of free Lévy laws related to a regression problem. J. Funct. Anal. 236 (2006), no. 1, 59--77. MR2227129
  • Bożejko, Marek; Lytvynov, Eugene. Meixner class of non-commutative generalized stochastic processes with freely independent values. I. A characterization. Comm. Math. Phys. 292 (2009), no. 1, 99--129. MR2540072
  • Bożejko, Marek; Lytvynov, Eugene. Meixner class of non-commutative generalized stochastic processes with freely independent values II. The generating function. Comm. Math. Phys. 302 (2011), no. 2, 425--451. MR2770019
  • Bożejko, Marek; Wysoczański, Janusz. Remarks on $t$-transformations of measures and convolutions. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 6, 737--761. MR1863276
  • Bryc, Wlodzimierz. Classical versions of $q$-Gaussian processes: conditional moments and Bell's inequality. Comm. Math. Phys. 219 (2001), no. 2, 259--270. MR1833804
  • Bryc, Włodzimierz. Free exponential families as kernel families. Demonstratio Math. 42 (2009), no. 3, 657--672. MR2554960
  • Bryc, Włodek. Markov processes with free-Meixner laws. Stochastic Process. Appl. 120 (2010), no. 8, 1393--1403. MR2653258
  • Chihara, T. S. An introduction to orthogonal polynomials. Mathematics and its Applications, Vol. 13. Gordon and Breach Science Publishers, New York-London-Paris, 1978. xii+249 pp. ISBN: 0-677-04150-0 MR0481884
  • Ejsmont, Wiktor. Laha-Lukacs properties of some free processes. Electron. Commun. Probab. 17 (2012), no. 13, 8 pp. MR2900354
  • Ejsmont W.: Characterizations of some free random variables by properties of conditional moments of third degree. To be published in Journal of Theoretical Probability, (2013).
  • Hiai, Fumio; Petz, Dénes. The semicircle law, free random variables and entropy. Mathematical Surveys and Monographs, 77. American Mathematical Society, Providence, RI, 2000. x+376 pp. ISBN: 0-8218-2081-8 MR1746976
  • Hora, Akihito; Obata, Nobuaki. Quantum probability and spectral analysis of graphs. With a foreword by Luigi Accardi. Theoretical and Mathematical Physics. Springer, Berlin, 2007. xviii+371 pp. ISBN: 978-3-540-48862-0 MR2316893
  • Michna, Z. Approximation of a symmetric $\alpha$-stable Lévy process by a Lévy process with finite moments of all orders. Studia Math. 180 (2007), no. 1, 1--10. MR2302007
  • Meixner, J. Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion. J. London Math. Soc. S1-9 no. 1, 6. MR1574715
  • Nica, Alexandru; Speicher, Roland. Lectures on the combinatorics of free probability. London Mathematical Society Lecture Note Series, 335. Cambridge University Press, Cambridge, 2006. xvi+417 pp. ISBN: 978-0-521-85852-6; 0-521-85852-6 MR2266879
  • Saitoh, Naoko; Yoshida, Hiroaki. The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory. Probab. Math. Statist. 21 (2001), no. 1, Acta Univ. Wratislav. No. 2298, 159--170. MR1869728
  • Speicher, Roland. Multiplicative functions on the lattice of noncrossing partitions and free convolution. Math. Ann. 298 (1994), no. 4, 611--628. MR1268597
  • Speicher, Roland. Free probability theory and non-crossing partitions. Sém. Lothar. Combin. 39 (1997), Art. B39c, 38 pp. (electronic). MR1490288
  • Szpojankowski K., Wesol owski J.: Dual lukacs regressions for non-commutative variables. arXiv:1110.3419v1 (2012).
  • Takesaki, Masamichi. Conditional expectations in von Neumann algebras. J. Functional Analysis 9 (1972), 306--321. MR0303307
  • Voiculescu, D. V.; Dykema, K. J.; 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, 1. American Mathematical Society, Providence, RI, 1992. vi+70 pp. ISBN: 0-8218-6999-X MR1217253

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.