IDEAS home Printed from https://ideas.repec.org/a/spr/sistpr/v23y2020i2d10.1007_s11203-020-09220-6.html
   My bibliography  Save this article

Asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion

Author

Listed:
  • Ciprian A. Tudor

    (Université de Lille 1)

  • Nakahiro Yoshida

    (University of Tokyo
    CREST, Japan Science and Technology Agency
    The Institute of Statistical Mathematics)

Abstract

We obtain the high-order asymptotic expansion for the distribution of the quadratic variation of the mixed fractional Brownian motion, which is defined as the sum of a Brownian motion and an independent fractional Brownian motion. Our approach is based on the analysis of the cumulants of this sequence. We show that both the Brownian and fractional Brownian parts contribute to the asymptotic expansion.

Suggested Citation

  • Ciprian A. Tudor & Nakahiro Yoshida, 2020. "Asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 435-463, July.
    • Handle: RePEc:spr:sistpr:v:23:y:2020:i:2:d:10.1007_s11203-020-09220-6
    DOI: 10.1007/s11203-020-09220-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11203-020-09220-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11203-020-09220-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Podolskij, Mark & Veliyev, Bezirgen & Yoshida, Nakahiro, 2017. "Edgeworth expansion for the pre-averaging estimator," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3558-3595.
    2. Mark Podolskij & Bezirgen Veliyev & Nakahiro Yoshida, 2018. "Edgeworth expansion for Euler approximation of continuous diffusion processes," CREATES Research Papers 2018-28, Department of Economics and Business Economics, Aarhus University.
    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.
    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. Yamagishi, Hayate & Yoshida, Nakahiro, 2023. "Order estimate of functionals related to fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 490-543.
    2. Yoshida, Nakahiro, 2023. "Asymptotic expansion and estimates of Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 176-248.
    3. Tudor, Ciprian A. & Yoshida, Nakahiro, 2023. "High order asymptotic expansion for Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 443-492.

    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. Mark Podolskij & Bezirgen Veliyev & Nakahiro Yoshida, 2018. "Edgeworth expansion for Euler approximation of continuous diffusion processes," CREATES Research Papers 2018-28, Department of Economics and Business Economics, Aarhus University.
    3. Nourdin, Ivan & Peccati, Giovanni & Podolskij, Mark, 2011. "Quantitative Breuer-Major theorems," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 793-812, April.
    4. 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.
    5. Surgailis, Donatas & Teyssière, Gilles & Vaiciulis, Marijus, 2008. "The increment ratio statistic," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 510-541, March.
    6. Bai, Shuyang & Taqqu, Murad S., 2019. "Sensitivity of the Hermite rank," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 822-840.
    7. Bai, Shuyang & Taqqu, Murad S. & Zhang, Ting, 2016. "A unified approach to self-normalized block sampling," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2465-2493.
    8. Yamagishi, Hayate & Yoshida, Nakahiro, 2023. "Order estimate of functionals related to fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 490-543.
    9. Marco Dozzi & Yuliya Mishura & Georgiy Shevchenko, 2015. "Asymptotic behavior of mixed power variations and statistical estimation in mixed models," Statistical Inference for Stochastic Processes, Springer, vol. 18(2), pages 151-175, July.
    10. Shuyang Bai & Murad S. Taqqu, 2013. "Multivariate Limit Theorems In The Context Of Long-Range Dependence," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(6), pages 717-743, November.
    11. Yoshida, Nakahiro, 2023. "Asymptotic expansion and estimates of Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 176-248.
    12. Miguel A. Arcones, 1999. "The Law of the Iterated Logarithm over a Stationary Gaussian Sequence of Random Vectors," Journal of Theoretical Probability, Springer, vol. 12(3), pages 615-641, July.
    13. Ivan Nourdin & David Nualart, 2010. "Central Limit Theorems for Multiple Skorokhod Integrals," Journal of Theoretical Probability, Springer, vol. 23(1), pages 39-64, March.
    14. Giraitis, Liudas & Robinson, Peter M. & Surgailis, Donatas, 1999. "Variance-type estimation of long memory," Stochastic Processes and their Applications, Elsevier, vol. 80(1), pages 1-24, March.
    15. 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.
    16. Jan Gairing & Peter Imkeller & Radomyra Shevchenko & Ciprian Tudor, 2020. "Hurst Index Estimation in Stochastic Differential Equations Driven by Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1691-1714, September.
    17. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2017. "Scaling transition for nonlinear random fields with long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2751-2779.
    18. Jun Yuan & Haowei Wang & Szu Hui Ng & Victor Nian, 2020. "Ship Emission Mitigation Strategies Choice Under Uncertainty," Energies, MDPI, vol. 13(9), pages 1-20, May.
    19. 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.
    20. Anh, V. V. & Leonenko, N. N., 1999. "Non-Gaussian scenarios for the heat equation with singular initial conditions," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 91-114, November.

    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:spr:sistpr:v:23:y:2020:i:2:d:10.1007_s11203-020-09220-6. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.