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Functional limit theorems for generalized variations of the fractional Brownian sheet

Author

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  • Mikko S. Pakkanen

    (CREATES and Department of Economics and Business, Aarhus University)

  • Anthony Réveillac

    (CEREMADE, Université Paris-Dauphine)

Abstract

We prove functional central and non-central limit theorems for generalized variations of the anisotropic d-parameter fractional Brownian sheet (fBs) for any natural number d. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and on the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application, we derive functional limit theorems for power variations of the fBs and discuss what is a proper way to interpolate them to ensure functional convergence.

Suggested Citation

  • 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.
    • Handle: RePEc:aah:create:2014-14
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    File URL: https://repec.econ.au.dk/repec/creates/rp/14/rp14_14.pdf
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    References listed on IDEAS

    as
    1. Pakkanen, Mikko S., 2014. "Limit theorems for power variations of ambit fields driven by white noise," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1942-1973.
    2. Nourdin, Ivan & Peccati, Giovanni & Podolskij, Mark, 2011. "Quantitative Breuer-Major theorems," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 793-812, April.
    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. 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.
    5. Maejima, Makoto & Tudor, Ciprian A., 2013. "On the distribution of the Rosenblatt process," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1490-1495.
    6. 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.
    7. Nualart, D. & Ortiz-Latorre, S., 2008. "Central limit theorems for multiple stochastic integrals and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 614-628, April.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Mark Podolskij, 2014. "Ambit fields: survey and new challenges," CREATES Research Papers 2014-51, Department of Economics and Business Economics, Aarhus University.

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    More about this item

    Keywords

    Fractional Brownian sheet; central limit theorem; non-central limit theorem; Hermite sheet; power variation; Malliavin calculus;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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