Electronic Journal of Probability

Central Limit Theorems and Quadratic Variations in Terms of Spectral Density

Hermine Biermé, Aline Bonami, and José R. Leon

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We give a new proof and provide new bounds for the speed of convergence in the Central Limit Theorem of Breuer Major on stationary Gaussian time series, which generalizes to particular triangular arrays. Our assumptions are given in terms of the spectral density of the time series. We then consider generalized quadratic variations of Gaussian fields with stationary increments under the assumption that their spectral density is asymptotically self-similar and prove Central Limit Theorems in this context.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 13, 362-395.

Accepted: 18 February 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G15: Gaussian processes 60G10: Stationary processes 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M15: Spectral analysis 62M40: Random fields; image analysis 60H07: Stochastic calculus of variations and the Malliavin calculus

Central limit theorem Gaussian stationary process spectral density periodogram quadratic variations fractional Brownian Motion

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Biermé, Hermine; Bonami, Aline; Leon, José R. Central Limit Theorems and Quadratic Variations in Terms of Spectral Density. Electron. J. Probab. 16 (2011), paper no. 13, 362--395. doi:10.1214/EJP.v16-862. https://projecteuclid.org/euclid.ejp/1464820182

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  • Arcones, Miguel A. Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 (1994), no. 4, 2242–2274.
  • Bardet, Jean-Marc; Doukhan, Paul; León, José Rafael. Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate. J. Time Ser. Anal. 29 (2008), no. 5, 906–945.
  • Bardet, Jean-Marc; Lang, Gabriel; Oppenheim, Georges; Philippe, Anne; Stoev, Stilian; Taqqu, Murad S. Semi-parametric estimation of the long-range dependence parameter: a survey. Theory and applications of long-range dependence, 557–577, Birkhäuser Boston, Boston, MA, 2003.
  • Barndorff-Nielsen, Ole E.; Corcuera, José Manuel; Podolskij, Mark. Power variation for Gaussian processes with stationary increments. Stochastic Process. Appl. 119 (2009), no. 6, 1845–1865.
  • Biermé, Hermine; Richard, Frédéric. Estimation of anisotropic Gaussian fields through Radon transform. ESAIM Probab. Stat. 12 (2008), 30–50 (electronic).
  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
  • Bonami, Aline; Estrade, Anne. Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9 (2003), no. 3, 215–236.
  • Chan, Grace; Wood, Andrew T. A. Increment-based estimators of fractal dimension for two-dimensional surface data. Statist. Sinica 10 (2000), no. 2, 343–376.
  • Chan, Grace; Wood, Andrew T. A. Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields. Ann. Statist. 32 (2004), no. 3, 1222–1260.
  • J.F. Coeurjolly. Inférence statistique pour les mouvements browniens fractionnaires et multifractionnaires. PhD thesis, University Joseph Fourier, 2000. 
  • Cramér, Harald; Leadbetter, M. R. Stationary and related stochastic processes. Sample function properties and their applications. Reprint of the 1967 original. Dover Publications, Inc., Mineola, NY, 2004. xiv+348 pp. ISBN: 0-486-43827-9
  • Doob, J. L. Stochastic processes. John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. viii+654 pp.
  • Ginovian, M. S. Nonparametric estimation of the spectrum of homogeneous Gaussian fields. (Russian) ; translated from Izv. Nats. Akad. Nauk Armenii Mat. 34 (1999), no. 2, 5–19 (2000) J. Contemp. Math. Anal. 34 (1999), no. 2, 1–15 (2000)
  • Hannan, E. J. Time series analysis. Methuen's Monographs on Applied Probability and Statistics. Methuen& Co., Ltd., London; John Wiley& Sons, Inc., New York 1960 viii+152 pp.
  • Chaos expansions, multiple Wiener-Itô integrals and their applications. Papers from the workshop held in Guanajuato, July 27-31, 1992. Edited by Christian Houdré and Victor Pérez-Abreu. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1994. xiv+377 pp. ISBN: 0-8493-8072-3
  • Istas, Jacques; Lang, Gabriel. Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 4, 407–436.
  • Kent, John T.; Wood, Andrew T. A. Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. Roy. Statist. Soc. Ser. B 59 (1997), no. 3, 679–699.
  • Lang, Gabriel; Roueff, François. Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates. Stat. Inference Stoch. Process. 4 (2001), no. 3, 283–306.
  • Lieb, Elliott H.; Loss, Michael. Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. xxii+346 pp. ISBN: 0-8218-2783-9
  • Mandelbrot, Benoit B.; Van Ness, John W. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 1968 422–437.
  • Nourdin, Ivan; Peccati, Giovanni. Stein's method on Wiener chaos. Probab. Theory Related Fields 145 (2009), no. 1-2, 75–118.
  • Nourdin, Ivan; Peccati, Giovanni. Stein's method meets Malliavin calculus: a short survey with new estimates, Recent development in stochastic dynamics and stochastic analysis, World Scientific, 2009.
  • Nourdin, I.; Peccati, G.; Podolskij M., M.. Quantitative Breuer-Major Theorems, to appear in Stochastic Process. Appl., 2010.
  • Nourdin, Ivan; Peccati, Giovanni; Reinert, Gesine. Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. Ann. Probab. 38 (2010), no. 5, 1947–1985.
  • Nourdin, Ivan; Peccati, Giovanni; Réveillac, Anthony. Multivariate normal approximation using Stein's method and Malliavin calculus. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 45–58. (Review)
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5
  • Nualart, D.; Ortiz-Latorre, S. Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 (2008), no. 4, 614–628.
  • Peccati, Giovanni; Tudor, Ciprian A. Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII, 247–262, Lecture Notes in Math., 1857, Springer, Berlin, 2005.
  • Yaglom, A. M. Correlation theory of stationary and related random functions. Vol. II. Supplementary notes and references. Springer Series in Statistics. Springer-Verlag, New York, 1987. x+258 pp. ISBN: 0-387-96331-6
  • Zhu, Zhengyuan; Stein, Michael L. Parameter estimation for fractional Brownian surfaces. Statist. Sinica 12 (2002), no. 3, 863–883.
  • Zygmund, A. Trigonometric series. Vol. I, II. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. xii; Vol. I: xiv+383 pp.; Vol. II: viii+364 pp. ISBN: 0-521-89053-5