IDEAS home Printed from https://ideas.repec.org/p/aah/create/2010-22.html
   My bibliography  Save this paper

Quantitative Breuer-Major Theorems

Author

Listed:
  • Ivan Nourdin

    (Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie)

  • Giovanni Peccati

    (University of Luxembourg)

  • Mark Podolskij

    (ETH Zürich and CREATES)

Abstract

We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n f(X_k)$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ is a measurable function. It is known that, under certain conditions on $f$ and the covariance function $r$ of $X$, $S_n$ converges in distribution to a normal variable $S$. In the present paper we derive several explicit upper bounds for quantities of the type $|\E[h(S_n)] -\E[h(S)]|$, 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 $\E[f^2(X_1)]$ and on simple infinite series involving the components of $r$. In particular, our results generalize and refine some classic CLTs by Breuer-Major, Giraitis-Surgailis and Arcones, concerning the normal approximation of partial sums associated with Gaussian-subordinated time-series.

Suggested Citation

  • Ivan Nourdin & Giovanni Peccati & Mark Podolskij, 2010. "Quantitative Breuer-Major Theorems," CREATES Research Papers 2010-22, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2010-22
    as

    Download full text from publisher

    File URL: https://repec.econ.au.dk/repec/creates/rp/10/rp10_22.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. Corinne Berzin & José León, 2007. "Estimating the Hurst Parameter," Statistical Inference for Stochastic Processes, Springer, vol. 10(1), pages 49-73, January.
    6. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. Mikkel Bennedsen & Ulrich Hounyo & Asger Lunde & Mikko S. Pakkanen, 2016. "The Local Fractional Bootstrap," Papers 1605.00868, arXiv.org, revised Oct 2017.
    3. 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.
    4. 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.
    5. 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.
    6. Marie Kratz & Sreekar Vadlamani, 2016. "CLT for Lipschitz-Killing curvatures of excursion sets of Gaussian random fields," Working Papers hal-01373091, HAL.
    7. 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.
    8. 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.
    9. 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).
    10. 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.
    11. Li, Yicun & Teng, Yuanyang, 2023. "Statistical inference in discretely observed fractional Ornstein–Uhlenbeck processes," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    12. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kerstin Gärtner & Mark Podolskij, 2014. "On non-standard limits of Brownian semi-stationary," CREATES Research Papers 2014-50, Department of Economics and Business Economics, Aarhus University.
    2. Andreas Basse-O'Connor & Raphaël Lachièze-Rey & Mark Podolskij, 2015. "Limit theorems for stationary increments Lévy driven moving averages," CREATES Research Papers 2015-56, Department of Economics and Business Economics, Aarhus University.
    3. 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.
    4. Mark Podolskij, 2014. "Ambit fields: survey and new challenges," CREATES Research Papers 2014-51, Department of Economics and Business Economics, Aarhus University.
    5. Mikkel Bennedsen & Ulrich Hounyo & Asger Lunde & Mikko S. Pakkanen, 2016. "The Local Fractional Bootstrap," Papers 1605.00868, arXiv.org, revised Oct 2017.
    6. 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.
    7. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij, 2009. "Limit theorems for functionals of higher order differences of Brownian semi-stationary processes," CREATES Research Papers 2009-60, Department of Economics and Business Economics, Aarhus University.
    8. Gärtner, Kerstin & Podolskij, Mark, 2015. "On non-standard limits of Brownian semi-stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 653-677.
    9. Mikkel Bennedsen, 2016. "Semiparametric inference on the fractal index of Gaussian and conditionally Gaussian time series data," Papers 1608.01895, arXiv.org, revised Mar 2018.
    10. Mark Podolskij & Katrin Wasmuth, 2012. "Goodness-of-fit testing for fractional diffusions," CREATES Research Papers 2012-12, Department of Economics and Business Economics, Aarhus University.
    11. Corcuera, José Manuel & Hedevang, Emil & Pakkanen, Mikko S. & Podolskij, Mark, 2013. "Asymptotic theory for Brownian semi-stationary processes with application to turbulence," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2552-2574.
    12. 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.
    13. Mikkel Bennedsen, 2016. "Semiparametric inference on the fractal index of Gaussian and conditionally Gaussian time series data," CREATES Research Papers 2016-21, Department of Economics and Business Economics, Aarhus University.
    14. José Manuel Corcuera, 2012. "New Central Limit Theorems for Functionals of Gaussian Processes and their Applications," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 477-500, September.
    15. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2017. "Decoupling the short- and long-term behavior of stochastic volatility," CREATES Research Papers 2017-26, Department of Economics and Business Economics, Aarhus University.
    16. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2016. "Decoupling the short- and long-term behavior of stochastic volatility," Papers 1610.00332, arXiv.org, revised Jan 2021.
    17. Debashis Mondal & Donald Percival, 2010. "Wavelet variance analysis for gappy time series," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(5), pages 943-966, October.
    18. Surgailis, Donatas & Teyssière, Gilles & Vaiciulis, Marijus, 2008. "The increment ratio statistic," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 510-541, March.
    19. Bai, Shuyang & Taqqu, Murad S., 2019. "Sensitivity of the Hermite rank," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 822-840.
    20. Andreas Basse-O'Connor & Mark Podolskij, 2015. "On critical cases in limit theory for stationary increments Lévy driven moving averages," CREATES Research Papers 2015-57, Department of Economics and Business Economics, Aarhus University.

    More about this item

    Keywords

    Berry-Esseen bounds; Breuer-Major central limit theorems; Gaussian processes; Interpolation; Malliavin calculus; Stein’s method;
    All these keywords.

    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

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:aah:create:2010-22. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: the person in charge (email available below). General contact details of provider: http://www.econ.au.dk/afn/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.