Strong mixing coefficients for non-commutative Gaussian processes
HTML articles powered by AMS MathViewer
- by Włodzimierz Bryc and Victor Kaftal PDF
- Proc. Amer. Math. Soc. 132 (2004), 523-534 Request permission
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
- István Berkes and Walter Philipp, Limit theorems for mixing sequences without rate assumptions, Ann. Probab. 26 (1998), no. 2, 805–831. MR 1626531, DOI 10.1214/aop/1022855651
- Philippe Biane, Free hypercontractivity, Comm. Math. Phys. 184 (1997), no. 2, 457–474. MR 1462754, DOI 10.1007/s002200050068
- Marek Bożejko, A $q$-deformed probability, Nelson’s inequality and central limit theorems, Nonlinear fields: classical, random, semiclassical (Karpacz, 1991) World Sci. Publ., River Edge, NJ, 1991, pp. 312–335. MR 1146011
- Marek Bożejko, Ultracontractivity and strong Sobolev inequality for $q$-Ornstein-Uhlenbeck semigroup $(-1<q<1)$, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), no. 2, 203–220. MR 1811255, DOI 10.1142/S0219025799000114
- Marek Bożejko, Burkhard Kümmerer, and Roland Speicher, $q$-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), no. 1, 129–154. MR 1463036, DOI 10.1007/s002200050084
- Marek Bożejko and Roland Speicher, An example of a generalized Brownian motion, Comm. Math. Phys. 137 (1991), no. 3, 519–531. MR 1105428, DOI 10.1007/BF02100275
- Richard C. Bradley, A remark on the central limit question for dependent random variables, J. Appl. Probab. 17 (1980), no. 1, 94–101. MR 557438, DOI 10.2307/3212927
- Richard C. Bradley, Every “lower psi-mixing” Markov chain is “interlaced rho-mixing”, Stochastic Process. Appl. 72 (1997), no. 2, 221–239. MR 1486554, DOI 10.1016/S0304-4149(97)00090-2
- Richard C. Bradley. Introduction to strong mixing conditions. Technical report, Indiana University, Bloomington, 2002. ISBN 1-58902-566-0.
- Wlodzimierz Bryc, Classical versions of $q$-Gaussian processes: conditional moments and Bell’s inequality, Comm. Math. Phys. 219 (2001), no. 2, 259–270. MR 1833804, DOI 10.1007/s002200100401
- U. Frisch and R. Bourret, Parastochastics, J. Mathematical Phys. 11 (1970), 364–390. MR 260352, DOI 10.1063/1.1665149
- I. A. Ibragimov and Ju. V. Linnik, Nezavisimye stalionarno svyazannye velichiny, Izdat. “Nauka”, Moscow, 1965 (Russian). MR 0202176
- 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 0322926
- Marius Iosifescu, Limit theorems for $\phi$-mixing sequences. A survey, Proceedings of the Fifth Conference on Probability Theory (Braşov, 1974) Editura Acad. R.S.R., Bucharest, 1977, pp. 51–57 (English, with Romanian summary). MR 0461624
- Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
- A. N. Kolmogorov and Ju. A. Rozanov, On a strong mixing condition for stationary Gaussian processes, Teor. Verojatnost. i Primenen. 5 (1960), 222–227 (Russian, with English summary). MR 0133175
- Arnaud Denjoy, Sur certaines séries de Taylor admettant leur cercle de convergence comme coupure essentielle, C. R. Acad. Sci. Paris 209 (1939), 373–374 (French). MR 50
- Magda Peligrad, An invariance principle for $\phi$-mixing sequences, Ann. Probab. 13 (1985), no. 4, 1304–1313. MR 806227
- D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253, DOI 10.1090/crmm/001
Additional Information
- Włodzimierz Bryc
- Affiliation: Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221–0025
- Email: Wlodzimierz.Bryc@UC.edu
- Victor Kaftal
- Affiliation: Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221–0025
- MR Author ID: 96695
- Email: Victor.Kaftal@UC.edu
- Received by editor(s): September 12, 2002
- Published electronically: June 5, 2003
- Communicated by: David R. Larson
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 523-534
- MSC (2000): Primary 81S05; Secondary 60E99
- DOI: https://doi.org/10.1090/S0002-9939-03-07051-5
- MathSciNet review: 2022378