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Quantitative Breuer-Major theorems

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  • Nourdin, Ivan
  • Peccati, Giovanni
  • Podolskij, Mark

Abstract

We consider sequences of random variables of the type , n>=1, where is a d-dimensional Gaussian process and is a measurable function. It is known that, under certain conditions on f and the covariance function r of X, Sn converges in distribution to a normal variable S. In the present paper we derive several explicit upper bounds for quantities of the type , where h is a sufficiently smooth test function. Our methods are based on Malliavin calculus, on interpolation techniques and on the Stein's method for normal approximation. The bounds deduced in our paper depend only on and on simple infinite series involving the components of r. In particular, our results generalize and refine some classic CLTs given by Breuer and Major, Giraitis and Surgailis, and Arcones, concerning the normal approximation of partial sums associated with Gaussian-subordinated time series.

Suggested Citation

  • Nourdin, Ivan & Peccati, Giovanni & Podolskij, Mark, 2011. "Quantitative Breuer-Major theorems," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 793-812, April.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:4:p:793-812
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    References listed on IDEAS

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    1. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    2. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.
    3. Lihong Wang, 2002. "Asymptotics of statistical estimates in stochastic programming problems with long-range dependent samples," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 55(1), pages 37-54, March.
    4. Corinne Berzin & José León, 2007. "Estimating the Hurst Parameter," Statistical Inference for Stochastic Processes, Springer, vol. 10(1), pages 49-73, January.
    5. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij, 2009. "Multipower Variation for Brownian Semistationary Processes," CREATES Research Papers 2009-21, Department of Economics and Business Economics, Aarhus University.
    6. León, José & Ludeña, Carenne, 2007. "Limits for weighted p-variations and likewise functionals of fractional diffusions with drift," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 271-296, March.
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    Cited by:

    1. Mikko S. Pakkanen & Anthony Réveillac, 2014. "Functional limit theorems for generalized variations of the fractional Brownian sheet," CREATES Research Papers 2014-14, Department of Economics and Business Economics, Aarhus University.
    2. Li, Yuan & Pakkanen, Mikko S. & Veraart, Almut E.D., 2023. "Limit theorems for the realised semicovariances of multivariate Brownian semistationary processes," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 202-231.
    3. Marie Kratz & Sreekar Vadlamani, 2018. "Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1729-1758, September.
    4. Pham, Viet-Hung, 2013. "On the rate of convergence for central limit theorems of sojourn times of Gaussian fields," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2158-2174.
    5. Mikkel Bennedsen & Ulrich Hounyo & Asger Lunde & Mikko S. Pakkanen, 2016. "The Local Fractional Bootstrap," CREATES Research Papers 2016-15, Department of Economics and Business Economics, Aarhus University.
    6. Bennedsen, Mikkel & Lunde, Asger & Shephard, Neil & Veraart, Almut E.D., 2023. "Inference and forecasting for continuous-time integer-valued trawl processes," Journal of Econometrics, Elsevier, vol. 236(2).
    7. Nourdin, Ivan & Nualart, David & Peccati, Giovanni, 2021. "The Breuer–Major theorem in total variation: Improved rates under minimal regularity," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 1-20.
    8. Marie Kratz & Sreekar Vadlamani, 2016. "CLT for Lipschitz-Killing curvatures of excursion sets of Gaussian random fields," Working Papers hal-01373091, HAL.
    9. Mikkel Bennedsen & Ulrich Hounyo & Asger Lunde & Mikko S. Pakkanen, 2016. "The Local Fractional Bootstrap," Papers 1605.00868, arXiv.org, revised Oct 2017.
    10. Bardet, Jean-Marc & Surgailis, Donatas, 2013. "Moment bounds and central limit theorems for Gaussian subordinated arrays," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 457-473.
    11. Nicolaescu, Liviu I., 2017. "A CLT concerning critical points of random functions on a Euclidean space," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3412-3446.

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    More about this item

    Keywords

    Berry-Esseen bounds Breuer-Major central limit theorems Gaussian processes Interpolation Malliavin calculus Stein's method;

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General

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