Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Strong mixing coefficients for non-commutative Gaussian processes

Authors: Wlodzimierz Bryc and Victor Kaftal
Journal: Proc. Amer. Math. Soc. 132 (2004), 523-534
MSC (2000): Primary 81S05; Secondary 60E99
Published electronically: June 5, 2003
MathSciNet review: 2022378
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Bounds for non-commutative versions of two classical strong mixing coefficients for $q$-Gaussian processes are found in terms of the angle between the underlying Hilbert spaces. As a consequence, we construct a $\psi$-mixing $q$-Gaussian stationary sequence with growth conditions on variances of partial sums. If classical processes with analogous properties were to exist, they would provide a counter-example to the Ibragimov conjecture.

References [Enhancements On Off] (What's this?)

  • 1. István Berkes and Walter Philipp.
    Limit theorems for mixing sequences without rate assumptions.
    Ann. Probab., 26(2):805-831, 1998. MR 99d:60022
  • 2. Philippe Biane.
    Free hypercontractivity.
    Comm. Math. Phys., 184(2):457-474, 1997. MR 98g:46097
  • 3. Marek Bozejko.
    A $q$-deformed probability, Nelson's inequality and central limit theorems.
    In Garbaczewski and Popowicz, editors, Nonlinear fields: classical, random, semiclassical (Karpacz, 1991), pages 312-335. World Sci. Publishing Co., River Edge, NJ, 1991. MR 93a:81098
  • 4. Marek Bozejko.
    Ultracontractivity and strong Sobolev inequality for $q$-Ornstein-Uhlenbeck semigroup ($-1<q<1$).
    Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2(2):203-220, 1999. MR 2002d:47058
  • 5. Marek Bozejko, Burkhard Kümmerer, and Roland Speicher.
    $q$-Gaussian processes: Non-commutative and classical aspects.
    Comm. Math. Physics, 185:129-154, 1997. MR 98h:81053
  • 6. Marek Bozejko and Roland Speicher.
    An example of a generalized Brownian motion.
    Comm. Math. Phys., 137(3):519-531, 1991. MR 92m:46096
  • 7. Richard C. Bradley.
    A remark on the central limit question for dependent random variables.
    J. Appl. Probab., 17(1):94-101, 1980. MR 82j:60031
  • 8. Richard C. Bradley.
    Every ``lower psi-mixing'' Markov chain is ``interlaced rho-mixing''.
    Stochastic Process. Appl., 72(2):221-239, 1997. MR 98m:60053
  • 9. Richard C. Bradley.
    Introduction to strong mixing conditions.
    Technical report, Indiana University, Bloomington, 2002.
    ISBN 1-58902-566-0.
  • 10. W\lodzimierz Bryc.
    Classical versions of $q$-Gaussian processes: conditional moments and Bell's inequality.
    Comm. Math. Physics, 219:259-270, 2001. MR 2002h:81129
  • 11. U. Frisch and R. Bourret.
    J. Mathematical Physics, 11(2):364-390, 1970. MR 41:4979
  • 12. I. A. Ibragimov and Ju. V. Linnik.
    Nezavisimye stalionarno svyazannye velichiny.
    Izdat. ``Nauka'', Moscow, 1965. MR 34:2049
  • 13. I. A. Ibragimov and Yu. V. Linnik.
    Independent and stationary sequences of random variables.
    Wolters-Noordhoff Publishing, Groningen, 1971.
    With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman. MR 48:1287
  • 14. Marius Iosifescu.
    Limit theorems for $\phi $-mixing sequences. A survey.
    In Proceedings of the Fifth Conference on Probability Theory (Brasov, 1974), pages 51-57. Editura Acad. R.S.R., Bucharest, 1977. MR 57:1609
  • 15. Richard V. Kadison and John R. Ringrose.
    Fundamentals of the theory of operator algebras.
    Interscience Publishers, Inc., Academic Press, 1983. MR 85j:46099
  • 16. A. N. Kolmogorov and Ju. A. Rozanov.
    On a strong mixing condition for stationary Gaussian processes.
    Teor. Verojatnost. i Primenen., 5:222-227, 1960. MR 24:A3009
  • 17. Edward Nelson.
    Notes on non-commutative integration.
    J. Functional Analysis, 15:103-116, 1974.MR 50:8102
  • 18. Magda Peligrad.
    An invariance principle for $\phi$-mixing sequences.
    Ann. Probab., 13(4):1304-1313, 1985. MR 87b:60056
  • 19. D. V. Voiculescu, K. J. Dykema, and A. Nica.
    Free random variables.
    American Mathematical Society, Providence, RI, 1992. MR 94c:46133

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 81S05, 60E99

Retrieve articles in all journals with MSC (2000): 81S05, 60E99

Additional Information

Wlodzimierz Bryc
Affiliation: Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221–0025

Victor Kaftal
Affiliation: Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221–0025

Keywords: Non-commutative uniform strong mixing, Ibragimov's conjecture, covariance estimates
Received by editor(s): September 12, 2002
Published electronically: June 5, 2003
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society